text: 4 TABLE I. Extracted simulation energies in lattice units for the variational and multiexponential ﬁtting methods. State Variational Multiexponential Ωb(1/2+) g.s. 0.7324(50)(24) 0.7262(43) Ωb(1/2+) excited 0.9938(278)(10) 0.9904(329) − Ωb(1/2 ) g.s. 0.8847(99)(90) 0.8801(44) Ωbb(1/2+) g.s. 0.6201(29)(20) 0.6137(22) Ωbb(1/2+) excited 0.8267(133)(24) 0.8015(139) Ω b -2 10 -4 10 -6 10 -8 10 0 4 8 12 16 20 t FIG. 4. Correlation functions for the positive parity Ω along with the result of a simultaneous four-term multiexponential ﬁt. b done using a diﬀerent method and a diﬀerent set of correlation functions. Since smearing at the sink usually leads to noisier correlators than smearing at the source a set of correlation function was constructed with source operators using combinations of ﬁelds gpg, gpe, epe, ggg, ege (6) where e denotes smearing with an excited state proﬁle and a local operator ppp at the sink. A simultaneous constrained multiexponential ﬁt [17] to the ﬁve correlation functions was done using four terms. As in the variational analysis correlation functions were averaged over a set of sixteen diﬀerent time sources for each gauge conﬁguration. The correlation functions and ﬁts are shown in ﬁg. 4 to ﬁg. 6. The time range used for ﬁtting was 2 to 18 for Ω and b 3 to 20 for Ω . The source corresponds to t = 1. The simulation energies for the ground and ﬁrst excited states are bb given in table I. They are compatible within statistical errors with the results obtained using the correlator matrix variational method. Since quark mass has been removed from the lattice NRQCD Hamiltonian the simulation energies extracted from the correlation do not give the hadron mass directly. However, diﬀerences between simulation energies in lattice units are related to mass diﬀerences by the inverse of the lattice spacing. Our calculated mass diﬀerences ∆M between the ﬁrst radial excitation and the ground state for Ω and for Ω are given in table II. The three errors shown for b bb this work are the statistical error and uncertainties due to ﬁtting time range and the lattice spacing determination. Table II also shows the calculated mass diﬀerence between the positive and negative parity ground states of Ω . b For comparison we note that an earlier lattice study [1] gave mass diﬀerences for Λ and Σ of 344(105)MeV and − − b b 252(60)MeV respectively. The PGD value for the Λb(1/2 ) Λb(1/2+) mass diﬀerence is 293(1)MeV . One way to get the baryon mass would be to calculate the kinetic energy as a function of momentum and determine a kinetic mass. However, this would be very time consuming and likely would have large statistical errors. Instead noitcnuf noitalerroC ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I.

text: Extracted

label: caption

text: simulation

text: energies

text: in

text: lattice

text: units

text: for

text: the

text: variational

text: and

text: multiexponential

text: ﬁtting

text: methods.

text: State

label: table

text: Variational

text: Multiexponential

text: g.s.

text: excited

text: g.s.

text: g.s.

text: excited

text: Ω

font: default

text: b

text: t

text: FIG.

text: Correlation

text: functions

text: for

text: the

text: positive

text: parity

text: Ω

text: along

text: with

text: the

text: result

text: of

text: a

text: simultaneous

text: four-term

text: multiexponential

text: ﬁt.

text: b

text: done

text: using

text: a

text: diﬀerent

text: method

text: and

text: a

text: diﬀerent

text: set

text: of

text: correlation

text: functions.

text: Since

text: smearing

text: at

text: the

text: sink

text: usually

text: leads

text: to

text: noisier

text: correlators

text: than

text: smearing

text: at

text: the

text: source

text: a

text: set

text: of

text: correlation

text: function

text: was

text: constructed

text: with

text: source

text: operators

text: using

text: combinations

text: of

text: ﬁelds

text: gpg,

text: gpe,

text: epe,

text: ggg,

text: ege

text: where

text: e

text: denotes

text: smearing

text: with

text: an

text: excited

text: state

text: proﬁle

text: and

text: a

text: local

text: operator

text: ppp

text: at

text: the

text: sink.

text: A

text: simultaneous

text: constrained

text: multiexponential

text: ﬁt

text: to

text: the

text: ﬁve

text: correlation

text: functions

text: was

text: done

text: using

text: four

text: terms.

text: As

text: in

text: the

text: variational

text: analysis

text: correlation

text: functions

text: were

text: averaged

text: over

text: a

text: set

text: of

text: sixteen

text: diﬀerent

text: time

text: sources

text: for

text: each

text: gauge

text: conﬁguration.

text: The

text: correlation

text: functions

text: and

text: ﬁts

text: are

text: shown

text: in

text: ﬁg.

text: to

text: ﬁg.

text: The

text: time

text: range

text: used

text: for

text: ﬁtting

text: was

text: to

text: for

text: Ω

text: and

text: b

text: to

text: for

text: Ω

text: The

text: source

text: corresponds

text: to

text: t

text: The

text: simulation

text: energies

text: for

text: the

text: ground

text: and

text: ﬁrst

text: excited

text: states

text: are

text: bb

text: given

text: in

text: table

text: I.

text: They

text: are

text: compatible

text: within

text: statistical

text: errors

text: with

text: the

text: results

text: obtained

text: using

text: the

text: correlator

text: matrix

text: variational

text: method.

text: Since

text: quark

text: mass

text: has

text: been

text: removed

text: from

text: the

text: lattice

text: NRQCD

text: Hamiltonian

text: the

text: simulation

text: energies

text: extracted

text: from

text: the

text: correlation

text: do

text: not

text: give

text: the

text: hadron

text: mass

text: directly.

text: However,

text: diﬀerences

text: between

text: simulation

text: energies

text: in

text: lattice

text: units

text: are

text: related

text: to

text: mass

text: diﬀerences

text: by

text: the

text: inverse

text: of

text: the

text: lattice

text: spacing.

text: Our

text: calculated

text: mass

text: diﬀerences

text: ∆M

text: between

text: the

text: ﬁrst

text: radial

text: excitation

text: and

text: the

text: ground

text: state

text: for

text: Ω

text: and

text: for

text: Ω

text: are

text: given

text: in

text: table

text: II.

text: The

text: three

text: errors

text: shown

text: for

text: b

text: bb

text: this

text: work

text: are

text: the

text: statistical

text: error

text: and

text: uncertainties

text: due

text: to

text: ﬁtting

text: time

text: range

text: and

text: the

text: lattice

text: spacing

text: determination.

text: Table

text: II

text: also

text: shows

text: the

text: calculated

text: mass

text: diﬀerence

text: between

text: the

text: positive

text: and

text: negative

text: parity

text: ground

text: states

text: of

text: Ω

text: b

text: For

text: comparison

text: we

text: note

text: that

text: an

text: earlier

text: lattice

text: study

text: gave

text: mass

text: diﬀerences

text: for

text: Λ

text: and

text: Σ

text: of

text: and

text: b

text: b

text: respectively.

text: The

text: PGD

text: value

text: for

text: the

text: mass

text: diﬀerence

text: is

text: One

text: way

text: to

text: get

text: the

text: baryon

text: mass

text: would

text: be

text: to

text: calculate

text: the

text: kinetic

text: energy

text: as

text: a

text: function

text: of

text: momentum

text: and

text: determine

text: a

text: kinetic

text: mass.

text: However,

text: this

text: would

text: be

text: very

text: time

text: consuming

text: and

text: likely

text: would

text: have

text: large

text: statistical

text: errors.

text: Instead

text: noitcnuf

text: noitalerroC

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text: ON AN ADDITIVE PRIME DIVISOR FUNCTION OF ALLADI AND ERDO˝ S DORIAN GOLDFELD This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday. (cid:80) Abstract. This paper discusses the additive prime divisor function A(n) := α p which || pα n was introduced by Alladi and Erd˝os in 1977. It is shown that A(n) is uniformly distributed (mod q) for any ﬁxed integer q > 1 with an explicit bound for the error. (cid:81) 1. Introduction r Let n = pai be the unique prime decomposition of a positive integer n. In 1977, Alladi i i=1 (cid:88) and Erd˝os [1] introduced the additive function r · A(n) := a p . i i i=1 Among several other things they proved that A(n) is uniformly distributed modulo 2. This (cid:88) was obtained from the identity ∞ − ( 1)A(n) 2s + 1 · ζ(2s) = − (1) ns 2s 1 ζ(s) n=1 together with the known z(cid:88)ero free region fo(cid:16)r the Riemann ze(cid:17)ta function. As a consequence they proved that there exists a constant c > 0 such that √ − O − ( 1)A(n) = x e c log xlog log x , ≤ n x → ∞ for x . In 1969 Delange [3] gave a necessary and suﬃcient condition for uniform distribution in progressions for integral valued additive functions which easily implies that A(n) is uniformly ≥ distributed (mod q) for all q 2 (although without a bound for the error in the asymptotic formula). The main goal of this paper is to show that A(n) is uniformly distributed modulo ≥ q for any integer q 2 with an explicit bound for the error. Unfortunately, it is not possible to ob(cid:88)tain such a simple identity as in (1) for the Dirichlet series ∞ e2πihAq(n) ns n=1 This research was partially supported by NSA grant H98230-15-1-0035. 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

label: paragraph

text: AN

text: ADDITIVE

text: PRIME

text: DIVISOR

text: FUNCTION

text: OF

text: ALLADI

text: AND

text: ERDO˝

text: S

text: DORIAN

text: GOLDFELD

text: This

text: paper

text: is

text: dedicated

text: to

text: Krishna

text: Alladi

text: on

text: the

text: occasion

text: of

text: his

text: birthday.

label: abstract

text: Abstract.

text: This

text: paper

text: discusses

text: the

text: additive

text: prime

text: divisor

text: function

text: A(n)

text: α

text: p

text: which

text: pα

text: n

text: was

text: introduced

text: by

text: Alladi

text: and

text: Erd˝os

text: in

text: It

text: is

text: shown

text: that

text: A(n)

text: is

text: uniformly

text: distributed

text: (mod

text: q)

text: for

text: any

text: ﬁxed

text: integer

text: q

text: with

text: an

text: explicit

text: bound

text: for

text: the

text: error.

label: section

text: Introduction

text: r

text: Let

text: n

text: pai

text: be

text: the

text: unique

text: prime

text: decomposition

text: of

text: a

text: positive

text: integer

text: n.

text: In

text: Alladi

text: i

text: and

text: Erd˝os

text: introduced

text: the

text: additive

text: function

text: r

text: A(n)

text: a

text: p

text: i

text: i

text: Among

text: several

text: other

text: things

text: they

text: proved

text: that

text: A(n)

text: is

text: uniformly

text: distributed

text: modulo

text: This

text: was

text: obtained

text: from

text: the

text: identity

text: ns

text: ζ(s)

text: together

text: with

text: the

text: known

text: free

text: region

text: the

text: Riemann

text: function.

text: As

text: a

text: consequence

text: they

text: proved

text: that

text: there

text: exists

text: a

text: constant

text: c

text: such

text: that

text: O

text: x

text: e

text: c

text: log

text: xlog

text: log

text: x

text: n

text: x

text: for

text: x

text: In

text: Delange

text: gave

text: a

text: necessary

text: and

text: suﬃcient

text: condition

text: for

text: uniform

text: distribution

text: in

text: progressions

text: for

text: integral

text: valued

text: additive

text: functions

text: which

text: easily

text: implies

text: that

text: A(n)

text: is

text: uniformly

text: distributed

text: (mod

text: q)

text: for

text: all

text: q

text: (although

text: without

text: a

text: bound

text: for

text: the

text: error

text: in

text: the

text: asymptotic

text: formula).

text: The

text: main

text: goal

text: of

text: this

text: paper

text: is

text: to

text: show

text: that

text: A(n)

text: is

text: uniformly

text: distributed

text: modulo

text: q

text: for

text: any

text: integer

text: q

text: with

text: an

text: explicit

text: bound

text: for

text: the

text: error.

text: Unfortunately,

text: it

text: is

text: not

text: possible

text: to

text: such

text: a

text: simple

text: identity

text: as

text: in

text: for

text: the

text: Dirichlet

text: series

text: ns

text: This

text: research

text: was

text: partially

text: supported

text: by

text: NSA

text: grant

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Hence, we may take (cid:88) (cid:88) ∞ 2πiphk − 2πipkh e q e q T (s) = , h,q k psk p k=2 (cid:60) (cid:3) which is easily seen to converge absolutely for (s) > 1. 2 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hence,

text: we

text: may

text: take

text: e

text: q

text: e

text: q

text: T

text: (s)

text: h,q

text: k

text: psk

text: p

text: which

text: is

text: easily

text: seen

text: to

text: converge

text: absolutely

text: for

text: (s)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [47] H. M. Sadjadi and M. Jamil, Europhys. Lett. 92, 69001 (2010). [48] N. Radicella and D. Pavon, Phys. Lett. B. 691, 121 (2010). [49] A. Sheykhi and M. Jamil, Gen. Rel. Grav. 43, 2661 (2011). [50] M. U. Farooq and M. Jamil, Canadian J. Phys. 89, 1251 (2011). [51] S. Das, S. Shankaranarayanan and S. Sur, Phys. Rev. D. 77, 064013 (2008). [52] G. Dvali and M. S. Turner, arXiv:astro-ph/0301510 (2003). [53] K. Karami, A. Abdolmaleki, N. Sahraei and S. Ghaﬀari, JHEP. 1108, 150 (2011). [54] A. Sheykhi and S. H. Hendi, Phys. Rev. D. 84, 044023 (2011). [55] J. J. Halliwell, Phys. Lett. B. 185, 341 (1987). [56] B. Ratra and P. J. E. Peebles, Phys. Rev. D. 37, 3406 (1988). [57] J. Yokoyama and K. I. Maeda, Phys. Lett. B. 207, 31 (1988). [58] A. A. Coley, J. Ibanez and R. J. van den Hoogen, J. Math. Phys. 38, 5256 (1997). [59] V. D. Ivashchuk, V. N. Melnikov, and A. B. Selivanov, JHEP. 0309, 059 (2003). [60] P. Brax, C van de Bruck, A. C. Davis, J. Khoury and A. Weltman, Phys. Rev. D. 70, 123518 (2004). [61] J. Khoury and A. Weltman, Phys. Rev. D. 69, 044026 (2004). [62] W. Fang, Y. Li, K. Zhang and H. Q. Lu, Class. Quant. Grav. 26, 155005 (2009). [63] K. Maeda and Y. Fujii, Phys. Rev. D. 79, 084026 (2009). [64] I. P. C. Heard and D. Wands, Class. Quant. Grav. 19, 5435-5448 (2002). [65] T. Barreiro, E. J. Copeland and N. J. Nunes, Phys. Rev. D. 61, 127301 (2000). [66] A. A. Sen and S. Sethi, Phys. Lett. B. 532, 159 (2002). [67] U. Franca and R. Rosenfeld, JHEP. 0210, 015 (2002). [68] R. Amanullah, et al., Astrophys. J. 716, 712738 (2010). 14

label: reference

text: H.

text: M.

text: Sadjadi

text: and

text: M.

text: Jamil,

text: Europhys.

text: Lett.

text: N.

text: Radicella

text: and

text: D.

text: Pavon,

text: Phys.

text: Lett.

text: B.

text: A.

text: Sheykhi

text: and

text: M.

text: Jamil,

text: Gen.

text: Rel.

text: Grav.

text: M.

text: U.

text: Farooq

text: and

text: M.

text: Jamil,

text: Canadian

text: J.

text: Phys.

text: S.

text: Das,

text: S.

text: Shankaranarayanan

text: and

text: S.

text: Sur,

text: Phys.

text: Rev.

text: D.

text: G.

text: Dvali

text: and

text: M.

text: S.

text: Turner,

text: K.

text: Karami,

text: A.

text: Abdolmaleki,

text: N.

text: Sahraei

text: and

text: S.

text: Ghaﬀari,

text: JHEP.

text: A.

text: Sheykhi

text: and

text: S.

text: H.

text: Hendi,

text: Phys.

text: Rev.

text: D.

text: J.

text: J.

text: Halliwell,

text: Phys.

text: Lett.

text: B.

text: B.

text: Ratra

text: and

text: P.

text: J.

text: E.

text: Peebles,

text: Phys.

text: Rev.

text: D.

text: J.

text: Yokoyama

text: and

text: K.

text: I.

text: Maeda,

text: Phys.

text: Lett.

text: B.

text: A.

text: A.

text: Coley,

text: J.

text: Ibanez

text: and

text: R.

text: J.

text: van

text: den

text: Hoogen,

text: J.

text: Math.

text: Phys.

text: V.

text: D.

text: Ivashchuk,

text: V.

text: N.

text: Melnikov,

text: and

text: A.

text: B.

text: Selivanov,

text: JHEP.

text: P.

text: Brax,

text: C

text: van

text: de

text: Bruck,

text: A.

text: C.

text: Davis,

text: J.

text: Khoury

text: and

text: A.

text: Weltman,

text: Phys.

text: Rev.

text: D.

text: J.

text: Khoury

text: and

text: A.

text: Weltman,

text: Phys.

text: Rev.

text: D.

text: W.

text: Fang,

text: Y.

text: Li,

text: K.

text: Zhang

text: and

text: H.

text: Q.

text: Lu,

text: Class.

text: Quant.

text: Grav.

text: K.

text: Maeda

text: and

text: Y.

text: Fujii,

text: Phys.

text: Rev.

text: D.

text: I.

text: P.

text: C.

text: Heard

text: and

text: D.

text: Wands,

text: Class.

text: Quant.

text: Grav.

text: T.

text: Barreiro,

text: E.

text: J.

text: Copeland

text: and

text: N.

text: J.

text: Nunes,

text: Phys.

text: Rev.

text: D.

text: A.

text: A.

text: Sen

text: and

text: S.

text: Sethi,

text: Phys.

text: Lett.

text: B.

text: U.

text: Franca

text: and

text: R.

text: Rosenfeld,

text: JHEP.

text: R.

text: Amanullah,

text: et

text: al.,

text: Astrophys.

text: J.


text: Long working distance objective lenses for single atom trapping and imaging Long working distance objective lenses for single atom trapping and imaginga) J. D. Pritchard,1,1, b) J. A. Isaacs,1 and M. Saﬀman1 (Dated: 29 May 2020) We present a pair of optimized objective lenses with long working distances of 117 mm and 65 mm respectively that oﬀer diﬀraction limited performance for both Cs and Rb wavelengths when imaging through standard vacuum windows. The designs utilise standard catalog lens elements to provide a simple and cost-eﬀective solution. Objective 1 provides NA = 0.175 oﬀering 3 µm resolution whilst objective 2 is optimized for high collection eﬃciency with NA = 0.29 and 1.8 µm resolution. This ﬂexible design can be further extended for use at shorter wavelengths by simply re-optimising the lens separations. PACS numbers: 42.15.Eq,32.50.+d Keywords: Imaging, Atom Trapping, Lens Design I. INTRODUCTION High resolution imaging plays a crucial role in atomic physics experiments, from trapping and imaging single atoms1–4 to creating complex arbitrary optical poten- tials for quantum gas studies5–7. These experiments are performed under vacuum requiring imaging through a glass window which introduces signiﬁcant spherical aber- rations and substantially reducing the resolution of an uncompensated optical system. One approach is to use lenses mounted inside the vacuum system such that col- FIG. 1. (Color online). ATOM objective, optimized for high limated light passes through the viewport8,9 however ad- NA ﬂuorescence detection for single atom readout at 852 nm. justment of the optical system is not possible once under vacuum and can additionally limit the lowest achievable pressure. provide inﬁnity-corrected and diﬀraction limited perfor- An alternative is to design a compound objective mance whilst maximising the available numerical aper- lens with the vacuum window included as the ﬁnal ele- ture given the minimum working distances imposed by ment as ﬁrst demonstrated by W. Alt10 which requires our vacuum chamber geometry. For simplicity all ele- four singlet lenses to provide diﬀraction limited perfor- ments are constrained to be 2 inch diameter BK7 singlets. mance. A number of similar designs have since been The objectives are iteratively designed using ray tracing published, either requiring one or more custom optics software (Zemax) to numerically optimise the lens spac- to be fabricated6,11 or using catalog optics but a short ing and curvatures to minimise the RMS wavefront error. focal length (35 mm) optimal for use with a small glass One by one each element is then replaced by the closest vacuum cell12. This precludes use with stainless steel matching catalog singlet and the remaining elements cur- vacuum systems that typically have much larger working vature and separation re-optimized until convergence is distances. obtained. Interestingly, the objective designs converge In this paper we present a pair of optimized objec- to the same lens shapes as used originally by W. Alt10 tive lenses with long working distances compatible for use suggesting this provides the optimal aberration compen- with a standard stainless steel chamber. These are de- sation when the ﬁnal element is a ﬂat glass window. The signed to be diﬀraction limited when used with standard objectives are then assembled in a standard 2 inch long conﬂat viewports and are comprized of standard catalog lens tube (Thorlabs SM2L02) using spacing rings ma- optics available with a range of anti-reﬂection coatings chined from delrin to avoid damaging the lenses. to provide a simple and cost-eﬀective solution. Whilst The ﬁrst objective (‘ATOM’) is designed to provide designed for use at 802 and 852 nm for trapping and high eﬃciency ﬂuorescence collection for imaging a sin- imaging neutral Cs atoms, the lenses remain diﬀraction gle Cs atom at 852 nm through a standard CF4.5” fused limited at 780 nm for use with Rb without modiﬁcation. silica viewport (thickness 6.4 mm) with a minimal work- For other elements, the lenses can be adapted to shorter ing distance of 65 mm, giving a maximum achievable or longer wavelengths by simply re-optimising the rela- numerical aperture 0.3. Figure 1 shows the ﬁnal lens tive spacings. conﬁguration along with spacing ring parameters, with II. DESIGN the complete prescription given in Table I. This lens has an eﬀective focal length of 67.4 mm and NA = 0.291 The objectives presented in this paper are designed to oﬀering a collection eﬃciency of 2.1 % and a diﬀraction ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Long

text: working

text: distance

text: objective

text: lenses

text: for

text: single

text: atom

text: trapping

text: and

text: imaging

text: Long

label: title

text: working

text: distance

text: objective

text: lenses

text: for

text: single

text: atom

text: trapping

text: and

text: imaginga)

text: J.

label: author

text: D.

text: b)

text: J.

text: A.

text: and

text: M.

text: (Dated:

label: date

text: May

text: We

text: present

text: a

text: pair

text: of

text: optimized

text: objective

text: lenses

text: with

text: long

text: working

text: distances

text: of

text: mm

text: and

text: mm

text: respectively

text: that

text: oﬀer

text: diﬀraction

text: limited

text: performance

text: for

text: both

text: Cs

text: and

text: Rb

text: wavelengths

text: when

text: imaging

text: through

text: standard

text: vacuum

text: windows.

text: The

text: designs

text: utilise

text: standard

text: catalog

text: lens

text: elements

text: to

text: provide

text: a

text: simple

text: and

text: cost-eﬀective

text: solution.

text: Objective

text: provides

text: oﬀering

text: µm

text: resolution

text: whilst

text: objective

text: is

text: optimized

text: for

text: high

text: collection

text: eﬃciency

text: with

text: and

text: µm

text: resolution.

text: This

text: ﬂexible

text: design

text: can

text: be

text: further

text: extended

text: for

text: use

text: at

text: shorter

text: wavelengths

text: by

text: simply

text: re-optimising

text: the

text: lens

text: separations.

text: PACS

text: numbers:

text: Keywords:

text: Imaging,

text: Atom

text: Trapping,

text: Lens

text: Design

text: I.

text: INTRODUCTION

text: High

text: resolution

text: imaging

text: plays

text: a

text: crucial

text: role

text: in

text: atomic

text: physics

text: experiments,

text: from

text: trapping

text: and

text: imaging

text: single

text: to

text: creating

text: complex

text: arbitrary

text: optical

text: poten-

text: tials

text: for

text: quantum

text: gas

text: These

text: experiments

text: are

text: performed

text: under

text: vacuum

text: requiring

text: imaging

text: through

text: a

text: glass

text: window

text: which

text: introduces

text: signiﬁcant

text: spherical

text: aber-

text: rations

text: and

text: substantially

text: reducing

text: the

text: resolution

text: of

text: an

text: uncompensated

text: optical

text: system.

text: One

text: approach

text: is

text: to

text: use

text: lenses

text: mounted

text: inside

text: the

text: vacuum

text: system

text: such

text: that

text: col-

text: FIG.

text: (Color

text: online).

text: ATOM

text: objective,

text: optimized

text: for

text: high

text: limated

text: light

text: passes

text: through

text: the

text: however

text: ad-

text: ﬂuorescence

text: detection

text: for

text: single

text: atom

text: readout

text: at

text: nm.

text: justment

text: of

text: the

text: optical

text: system

text: is

text: not

text: possible

text: once

text: under

text: vacuum

text: and

text: can

text: additionally

text: limit

text: the

text: lowest

text: achievable

text: pressure.

text: provide

text: inﬁnity-corrected

text: and

text: diﬀraction

text: limited

text: perfor-

text: An

text: alternative

text: is

text: to

text: design

text: a

text: compound

text: objective

text: mance

text: whilst

text: maximising

text: the

text: available

text: numerical

text: aper-

text: lens

text: with

text: the

text: vacuum

text: window

text: included

text: as

text: the

text: ﬁnal

text: ele-

text: ture

text: given

text: the

text: minimum

text: working

text: distances

text: imposed

text: by

text: ment

text: as

text: ﬁrst

text: demonstrated

text: by

text: W.

text: which

text: requires

text: our

text: vacuum

text: chamber

text: geometry.

text: For

text: simplicity

text: all

text: ele-

text: four

text: singlet

text: lenses

text: to

text: provide

text: diﬀraction

text: limited

text: perfor-

text: ments

text: are

text: constrained

text: to

text: be

text: inch

text: diameter

text: singlets.

text: mance.

text: A

text: number

text: of

text: similar

text: designs

text: have

text: since

text: been

text: The

text: objectives

text: are

text: iteratively

text: designed

text: using

text: ray

text: tracing

text: published,

text: either

text: requiring

text: one

text: or

text: more

text: custom

text: optics

text: software

text: (Zemax)

text: to

text: numerically

text: optimise

text: the

text: lens

text: spac-

text: to

text: be

text: or

text: using

text: catalog

text: optics

text: but

text: a

text: short

text: ing

text: and

text: curvatures

text: to

text: minimise

text: the

text: RMS

text: wavefront

text: error.

text: focal

text: length

text: mm)

text: optimal

text: for

text: use

text: with

text: a

text: small

text: glass

text: One

text: by

text: one

text: each

text: element

text: is

text: then

text: replaced

text: by

text: the

text: closest

text: vacuum

text: This

text: precludes

text: use

text: with

text: stainless

text: steel

text: matching

text: catalog

text: singlet

text: and

text: the

text: remaining

text: elements

text: cur-

text: vacuum

text: systems

text: that

text: typically

text: have

text: much

text: larger

text: working

text: vature

text: and

text: separation

text: re-optimized

text: until

text: convergence

text: is

text: distances.

text: obtained.

text: Interestingly,

text: the

text: objective

text: designs

text: converge

text: In

text: this

text: paper

text: we

text: present

text: a

text: pair

text: of

text: optimized

text: objec-

text: to

text: the

text: same

text: lens

text: shapes

text: as

text: used

text: originally

text: by

text: W.

text: tive

text: lenses

text: with

text: long

text: working

text: distances

text: compatible

text: for

text: use

text: suggesting

text: this

text: provides

text: the

text: optimal

text: aberration

text: compen-

text: with

text: a

text: standard

text: stainless

text: steel

text: chamber.

text: These

text: are

text: de-

text: sation

text: when

text: the

text: ﬁnal

text: element

text: is

text: a

text: ﬂat

text: glass

text: window.

text: The

text: signed

text: to

text: be

text: diﬀraction

text: limited

text: when

text: used

text: with

text: standard

text: objectives

text: are

text: then

text: assembled

text: in

text: a

text: standard

text: inch

text: long

text: conﬂat

text: viewports

text: and

text: are

text: comprized

text: of

text: standard

text: catalog

text: lens

text: tube

text: (Thorlabs

text: using

text: spacing

text: rings

text: ma-

text: optics

text: available

text: with

text: a

text: range

text: of

text: anti-reﬂection

text: coatings

text: chined

text: from

text: delrin

text: to

text: avoid

text: damaging

text: the

text: lenses.

text: to

text: provide

text: a

text: simple

text: and

text: cost-eﬀective

text: solution.

text: Whilst

text: The

text: ﬁrst

text: objective

text: (‘ATOM’)

text: is

text: designed

text: to

text: provide

text: designed

text: for

text: use

text: at

text: and

text: nm

text: for

text: trapping

text: and

text: high

text: eﬃciency

text: ﬂuorescence

text: collection

text: for

text: imaging

text: a

text: sin-

text: imaging

text: neutral

text: Cs

text: atoms,

text: the

text: lenses

text: remain

text: diﬀraction

text: gle

text: Cs

text: atom

text: at

text: nm

text: through

text: a

text: standard

text: fused

text: limited

text: at

text: nm

text: for

text: use

text: with

text: Rb

text: without

text: modiﬁcation.

text: silica

text: viewport

text: (thickness

text: mm)

text: with

text: a

text: minimal

text: work-

text: For

text: other

text: elements,

text: the

text: lenses

text: can

text: be

text: adapted

text: to

text: shorter

text: ing

text: distance

text: of

text: mm,

text: giving

text: a

text: maximum

text: achievable

text: or

text: longer

text: wavelengths

text: by

text: simply

text: re-optimising

text: the

text: rela-

text: numerical

text: aperture

text: Figure

text: shows

text: the

text: ﬁnal

text: lens

text: tive

text: spacings.

text: conﬁguration

text: along

text: with

text: spacing

text: ring

text: parameters,

text: with

text: II.

text: DESIGN

text: the

text: complete

text: prescription

text: given

text: in

text: Table

text: I.

text: This

text: lens

text: has

text: an

text: eﬀective

text: focal

text: length

text: of

text: mm

text: and

text: The

text: objectives

text: presented

text: in

text: this

text: paper

text: are

text: designed

text: to

text: oﬀering

text: a

text: collection

text: eﬃciency

text: of

text: and

text: a

text: diﬀraction

text: ##LTFigure##

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text: √ − (cid:15) 1 c2 HΩ = . (10) Λ r c By diﬀerentiating of the above equation with respect to t we yield √ − − (cid:15) 1 c2Ω˙ ˙ Λ H = . (11) r Ω2 c Λ On the other hand, the Gibbons-Hawking entropy is proportional to the squared radius of the universe that is taken as the Hubble horizon, i.e. ∼ ∼ ∼ − S A L2 H 2. (12) ≤ ˙ Since entropy should increase during expansion phase, we obtain H 0. From (11) and by assuming that Ω increases with time we again ﬁnd (cid:15) = +1. We should notice that the Λ two constraints on c and (cid:15) are the characteristics for this model only if Hubble radius is − − − taken as the horizon. By using Ω = 1 Ω = 1 Ω H2H 2(1 + z)3 and Ω = 1/(4r2H2) Λ m m0 0 rc c 0 in (10), the Friedmann equation becomes (cid:112) − E2 = 2 Ω (1 c2)E + Ω (1 + z)3. (13) rc m0 where we have used the deﬁnition E2 = H2/H2. This equation at z = 0 yields the 0 constraint (cid:112) − − 1 Ω = 2 Ω (1 c2). (14) m0 rc In the next section we are going to constraint the model parameters c, Ω and Ω with m0 rc the observational data by employing χ2 method. However, with attention to constraint (14) we implement this in two diﬀerent ways. A: Using (14) we can neglect c and Ω in favor of Ω in (13) and obtain rc m0 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: HΩ

text: Λ

text: r

text: c

text: By

text: diﬀerentiating

text: of

text: the

text: above

text: equation

text: with

text: respect

text: to

text: t

text: we

text: yield

text: c2Ω˙

text: Λ

text: H

text: r

text: Ω2

text: c

text: Λ

text: On

text: the

text: other

text: hand,

text: the

text: Gibbons-Hawking

text: entropy

text: is

text: proportional

text: to

text: the

text: squared

text: radius

text: of

text: the

text: universe

text: that

text: is

text: taken

text: as

text: the

text: Hubble

text: horizon,

text: i.e.

text: S

text: A

text: H

text: Since

text: entropy

text: should

text: increase

text: during

text: expansion

text: phase,

text: we

text: obtain

text: H

text: From

text: and

text: by

text: assuming

text: that

text: Ω

text: increases

text: with

text: time

text: we

text: again

text: ﬁnd

text: We

text: should

text: notice

text: that

text: the

text: Λ

text: two

text: constraints

text: on

text: c

text: and

text: are

text: the

text: characteristics

text: for

text: this

text: model

text: only

text: if

text: Hubble

text: radius

text: is

text: taken

text: as

text: the

text: horizon.

text: By

text: using

text: Ω

text: Ω

text: Ω

text: and

text: Ω

text: Λ

text: m

text: rc

text: c

text: in

text: the

text: Friedmann

text: equation

text: becomes

text: Ω

text: Ω

text: rc

text: where

text: we

text: have

text: used

text: the

text: deﬁnition

text: This

text: equation

text: at

text: z

text: yields

text: the

text: constraint

text: Ω

text: Ω

text: rc

text: In

text: the

text: next

text: section

text: we

text: are

text: going

text: to

text: constraint

text: the

text: model

text: parameters

text: c,

text: Ω

text: and

text: Ω

text: with

text: rc

text: the

text: observational

text: data

text: by

text: employing

text: χ2

text: method.

text: However,

text: with

text: attention

text: to

text: constraint

text: we

text: implement

text: this

text: in

text: two

text: diﬀerent

text: ways.

text: A:

text: Using

text: we

text: can

text: neglect

text: c

text: and

text: Ω

text: in

text: favor

text: of

text: Ω

text: in

text: and

text: obtain

text: rc

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: | | | | | | | | Let a := A /n, b := B /n, c := C /n and d := D /n. The edge-density of G(cid:48) can be upper- bounded by f(a, b, c, d) := 2ab + 2bc + 2cd + d2. Let us analyze the maximum value of f under constraints on a, b, c and d that we have already established. All of that is summarized in the following optimization problem: maximize: 2ab + 2bc + 2cd + d2 ≥ ≥ ≥ subject to: a 0(cid:16), b 0(cid:17), c 0, − − − d = 1 a b c, √ ≥ − ab 2 2 /16, ≥ d δ /2. 2 Clearly, if the values of a, b, c and d are equal to those coming from Construction 2, then ∈ R f(a, b, c, d) = 1/2. The following proposition shows that there is no other point (a(cid:48), b(cid:48), c(cid:48), d(cid:48)) 4 that would satisfy the constraints and also attain the value 1/2. (cid:32) (cid:33) Claim 5.15. The optimization problem has a unique solution at the point √ √ 1 − 2 1 1 2 (a , b , c , d ) = , , , , m m m m 2 4 4 4 4 Moreover, if a point (a(cid:48), b(cid:48), c(cid:48), d(cid:48)) satisﬁes all the constraints and f(a(cid:48), b(cid:48), c(cid:48), d(cid:48)) is close to 1/2, then (a(cid:48), b(cid:48), c(cid:48), d(cid:48)) is close to (a , b , c , d ). m m m m This claim immediately yields that G(cid:48) is close to Construction 2, hence proving the claim ﬁnishes the proof of Theorem 1.6 (cid:0) (cid:1) (cid:0) (cid:1) Proof of Claim 5.15. Let (a , b , c , d ) ∈ R4 be a point th√at satisﬁes the constraints and m√aximizes 0 0 0 0 − − the objective function. First, we show that a b = 2 2 /16. Indeed, if a b > 2 2 /16, 0 0 0 0 − √ then let α := a 2− 2. It follows that 0 16·b0 − · f(a α, b , c + α, d ) = f (a , b , c , d ) + α d , 0 0 0 0 0 0 0 0 0 and the point (a , b , c , d ) was not an optimal solution. 0 0 0 0 We continue by bounding b away from 1/2. Since 0 (cid:16) − − −(cid:17) − − − − f (a, b, c, d) = 2ab + 2(b + d)(1 a b d) + d2 = 2b(1 b) d2 + 2d (1 a 2b) , − − − √ − it holds that (1 a 2b ) = 1 2− 2 2b > 0. Therefore, 0 0 16·b0 0 2 + 23/4 b < < 0.47. (cid:16) (cid:17) 0 8 − Also, it follows that the maximum value of 2b(1 b) is less than 0.499. On the other hand, the − √ − maximum value of 1 2− 2 2b is less than 1/2, hence d > 0.001. 16·b0 0 0 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: a

text: A

text: /n,

text: b

text: B

text: /n,

text: c

text: C

text: /n

text: and

text: d

text: D

text: /n.

text: The

text: edge-density

text: of

text: can

text: be

text: upper-

text: bounded

text: by

text: f(a,

text: b,

text: c,

text: d)

text: Let

text: us

text: analyze

text: the

text: maximum

text: value

text: of

text: f

text: under

text: constraints

text: on

text: a,

text: b,

text: c

text: and

text: d

text: that

text: we

text: have

text: already

text: established.

text: All

text: of

text: that

text: is

text: summarized

text: in

text: the

text: following

text: optimization

text: problem:

text: maximize:

text: subject

text: to:

text: a

text: b

text: c

text: d

text: a

text: b

text: c,

text: ab

text: d

text: δ

text: Clearly,

text: if

text: the

text: values

text: of

text: a,

text: b,

text: c

text: and

text: d

text: are

text: equal

text: to

text: those

text: coming

text: from

text: Construction

text: then

text: R

text: f(a,

text: b,

text: c,

text: d)

text: The

text: following

text: proposition

text: shows

text: that

text: there

text: is

text: no

text: other

text: point

text: that

text: would

text: satisfy

text: the

text: constraints

text: and

text: also

text: attain

text: the

text: value

text: Claim

text: The

text: optimization

text: problem

text: has

text: a

text: unique

text: solution

text: at

text: the

text: point

text: (a

text: b

text: c

text: d

text: m

text: m

text: m

text: m

text: Moreover,

text: if

text: a

text: point

text: satisﬁes

text: all

text: the

text: constraints

text: and

text: is

text: close

text: to

text: then

text: is

text: close

text: to

text: (a

text: b

text: c

text: d

text: m

text: m

text: m

text: m

text: This

text: claim

text: immediately

text: yields

text: that

text: is

text: close

text: to

text: Construction

text: hence

text: proving

text: the

text: claim

text: ﬁnishes

text: the

text: proof

text: of

text: Theorem

text: Proof

text: of

text: Claim

text: Let

text: (a

text: b

text: c

text: d

text: be

text: a

text: point

text: th√at

text: satisﬁes

text: the

text: constraints

text: and

text: m√aximizes

text: the

text: objective

text: function.

text: First,

text: we

text: show

text: that

text: a

text: b

text: Indeed,

text: if

text: a

text: b

text: then

text: let

text: α

text: a

text: 2−

text: It

text: follows

text: that

text: f(a

text: α,

text: b

text: c

text: α,

text: d

text: f

text: (a

text: b

text: c

text: d

text: α

text: d

text: and

text: the

text: point

text: (a

text: b

text: c

text: d

text: was

text: not

text: an

text: optimal

text: solution.

text: We

text: continue

text: by

text: bounding

text: b

text: away

text: from

text: Since

text: f

text: (a,

text: b,

text: c,

text: d)

text: a

text: b

text: d)

text: b)

text: a

text: it

text: holds

text: that

text: a

text: 2−

text: Therefore,

text: b

text: Also,

text: it

text: follows

text: that

text: the

text: maximum

text: value

text: of

text: b)

text: is

text: less

text: than

text: On

text: the

text: other

text: hand,

text: the

text: maximum

text: value

text: of

text: 2−

text: is

text: less

text: than

text: hence

text: d

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • | | | | | | | | the quadruple ( A , B , C , D ) is n-extremal, • the A, B and C are independent sets of G, • { } ∈ ∈ ∈ u, v E(G) for any u A and v B, • { } ∈ ∈ ∈ ∪ u, v / E(G) for any u A and v C D, and • { } ∈ ∈ ∈ u, v / E(G) for any u B and v D. An immediate consequence of this theorem is that (a, b, c, d) is n-extremal if and only if it solves the following integer quadratic program: · maximize: a b (cid:18) (cid:19) ∈ N ∈ N ∈ N ∈ N subject to: a , b , c , d , · · · d n2 a b + b c + c d + > , 2 4 a + b + c + d = n. √ Since the exact solution of this maximization problem for a given integer n depends on errors in rounding expressions like 2n/4, we leave it in this form. Approximate values of a, b, c and d are indeed given by Construction 2. Proof of Theorem 6.1. Theorems 1.3 and 1.6 immediately yield that for any ε > 0, there exists an ≥ ∈ G integer n so that if n n , then by adding or(cid:101)removing εn2 edges in a graph G we obtain the 0 0 n √ √ graph from Construction 2. Moreover, the value of n will be large enough so that Construction 2 ± 0 − ± ≥ yields that F (n) = ((2 + 2)/16 ε)n2 and F (n) = ((2 2)/16 ε)n2 for every n n . ≥ ∈ G 0 Fix an integer n n and any graph G , and let V be the vertex-set of G. Clearly, for any 0 n ε(cid:48) > 0, we can ﬁnd ε > 0 small enough so that V can be partitioned into ﬁve sets A , B , C , D 0 0 0 0 and X such that √ √ • | | − ± · | | ± · | | ± · | | ± · A = (1/2 2/4 ε(cid:48)) n, B = (1/4 ε(cid:48)) n, C = (1/4 ε(cid:48)) n, D = ( 2/4 ε(cid:48)) n, 0 0 0 0 • ≤ | | ≤ · 0 X ε(cid:48) n, • ≥ − | | ∈ deg (u) (1 ε(cid:48)) A for every u B , A0 0 0 • ≥ − | | ∈ ∪ deg (u) (1 ε(cid:48)) B for every u A C , B0 0 0 0 • ≥ − | | ∈ ∪ deg (u) (1 ε(cid:48)) C for every u B D , C0 0 0 0 • ≥ − | | ∈ deg (u) (1 ε(cid:48)) D for every u C , and D0 0 0 • − the induced subgraph G[D ] has edge-density at least 1 ε(cid:48). 0 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: the

text: quadruple

text: A

text: B

text: C

text: D

text: is

text: n-extremal,

text: the

text: A,

text: B

text: and

text: C

text: are

text: independent

text: sets

text: of

text: G,

text: u,

text: v

text: E(G)

text: for

text: any

text: u

text: A

text: and

text: v

text: B,

text: u,

text: v

text: E(G)

text: for

text: any

text: u

text: A

text: and

text: v

text: C

text: D,

text: and

text: u,

text: v

text: E(G)

text: for

text: any

text: u

text: B

text: and

text: v

text: D.

text: An

text: immediate

text: consequence

text: of

text: this

text: theorem

text: is

text: that

text: (a,

text: b,

text: c,

text: d)

text: is

text: n-extremal

text: if

text: and

text: only

text: if

text: it

text: solves

text: the

text: following

text: integer

text: quadratic

text: program:

text: maximize:

text: a

text: b

text: N

text: N

text: N

text: N

text: subject

text: to:

text: a

text: b

text: c

text: d

text: d

text: a

text: b

text: b

text: c

text: c

text: d

text: a

text: b

text: c

text: d

text: n.

text: Since

text: the

text: exact

text: solution

text: of

text: this

text: maximization

text: problem

text: for

text: a

text: given

text: integer

text: n

text: depends

text: on

text: errors

text: in

text: rounding

text: expressions

text: like

text: we

text: leave

text: it

text: in

text: this

text: form.

text: Approximate

text: values

text: of

text: a,

text: b,

text: c

text: and

text: d

text: are

text: indeed

text: given

text: by

text: Construction

text: Proof

text: of

text: Theorem

text: Theorems

text: and

text: immediately

text: yield

text: that

text: for

text: any

text: ε

text: there

text: exists

text: an

text: G

text: integer

text: n

text: so

text: that

text: if

text: n

text: n

text: then

text: by

text: adding

text: edges

text: in

text: a

text: graph

text: G

text: we

text: obtain

text: the

text: n

text: graph

text: from

text: Construction

text: Moreover,

text: the

text: value

text: of

text: n

text: will

text: be

text: large

text: enough

text: so

text: that

text: Construction

text: yields

text: that

text: F

text: (n)

text: and

text: F

text: (n)

text: for

text: every

text: n

text: n

text: G

text: Fix

text: an

text: integer

text: n

text: n

text: and

text: any

text: graph

text: G

text: and

text: let

text: V

text: be

text: the

text: vertex-set

text: of

text: G.

text: Clearly,

text: for

text: any

text: n

text: we

text: can

text: ﬁnd

text: ε

text: small

text: enough

text: so

text: that

text: V

text: can

text: be

text: partitioned

text: into

text: ﬁve

text: sets

text: A

text: B

text: C

text: D

text: and

text: X

text: such

text: that

text: A

text: n,

text: B

text: n,

text: C

text: n,

text: D

text: n,

text: X

text: n,

text: deg

text: (u)

text: A

text: for

text: every

text: u

text: B

text: deg

text: (u)

text: B

text: for

text: every

text: u

text: A

text: C

text: deg

text: (u)

text: C

text: for

text: every

text: u

text: B

text: D

text: deg

text: (u)

text: D

text: for

text: every

text: u

text: C

text: and

text: the

text: induced

text: subgraph

text: G[D

text: has

text: edge-density

text: at

text: least

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∪ (cid:12) (cid:12) Let B := B B . Recall the deﬁnition of C , i.e., 0 1 1 { ∈ ≥ ∧ ∃ ∈ ∈ { } ∈ ∧ { } ∈ } C = u Z deg (u) 1 v V, w D : u, v E(G) v, w E(G) . 1 B0 0 By Claim 6.8, there are no edges between C and C , and by the deﬁnition of B , there are no 1 0 ∈ 1 edges between C and A. Let us now show that vertices v C have many neighbors in B . 1 1 0 ∈ ≥ Claim 6.19. For every u C , deg (u) n/25. 1 B0 ∪ ∪ Proof. As noted above, u can be adjacent only to the vertices in B D Z. First observe that ∈ { } ∈ C ∈ 0 0 ∈ for every z N (u), the edge u, z (G). Indeed, let x N (u) and y N (x) be chosen D0 ∈ ∈ 5 B0 C0 arbitrarily. Since y C and z D , there exist a common neighbor of y and z which encloses a 0 { } ∈ C0 ∈ ∈ C . Analogously, we show u, x (G) for every x N (u). Consider the vertices v V and 5∈ { } ∈ { 5} ∈ B0 ∈ ∈ ∈ w D with u, v E(G) and v, w E(G) witnessing that u C . Since x B and w D , 0 { } ∈ 1C 0 0 the two vertices must have a common neighbor which yields u, x (G). | | 5 The last paragraph shows that u√is incident to at most Z < 22ε(cid:48)n edges that do not occur in | ∪ | C . On the other hand, Z D < ( 2/4 + 23ε(cid:48))n. So if deg (v) < n/25, then 5 0 B0 deg(v) < 0.396n < deg(u(cid:48)), ∈ where u(cid:48) B is the vertex from Claim 6.11. Therefore, the graph obtained by removing the vertex 0 u and cloning the vertex u(cid:48) contradicts Claim 6.2. Corollary 6.20. C is an independent set in G. 1 { (cid:16) } (cid:17) ∈ Proof. Suppose for a contradiction(cid:0)there is an(cid:1)edge u, u(cid:48) with u, u(cid:48) C . There are at least 1 · | | − n · n − n2 deg (u) A ε(cid:48)n > 23ε(cid:48)n > B0 0 25 4 101 { } ∈ ∈ edges v, w with v N (u) and w N (v). However, the vertices w and u(cid:48) have a common \ { } B0 | A∩0 C | neighbor in B v and hence E(A , B ) (G) > n2/101; a contradiction with Corollary 6.6. 0 0 0 5 ∪ \ ∪ ∪ We deﬁne C := C C , and D := Z (A B C ). By the deﬁnition of the sets A, B and ∈ 0 1 1 1∪ 1 1 1 C , every vertex v D has no neigbors in A B . We now concentrate on the edges between D 1 1 0 1 and C . 0 ∈ ≥ Claim 6.21. For every v D , deg (v) n/25. 1 C0 ∪ ∪ Proof. The vertex v can be adjacent only to the vertices in C D Z, and clearly every edge { } ∈ ∪ 0 0 v, w with w C D occurs in C . In particular, v is incident to at most 22ε(cid:48)n edges that do 0 0 5 not occur in C . 5 | | | | As in Claim 6.19, if deg (v) < n/25 then deg(v) < D + Z + n/25 < 0.396n. Therefore, C0 0 removing the vertex v and cloning the vertex u(cid:48) from Claim 6.11 result in a graph contradicting Claim 6.2. So the only possible edges that could be in G but not following the pattern of Construction 2 are those between B and D . We rule them out in the following claim. 1 1 | | Claim 6.22. E(B , D ) = 0. 1 1 25 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: B

text: B

text: B

text: Recall

text: the

text: deﬁnition

text: of

text: C

text: i.e.,

text: C

text: u

text: Z

text: deg

text: (u)

text: v

text: V,

text: w

text: D

text: u,

text: v

text: E(G)

text: v,

text: w

text: E(G)

text: By

text: Claim

text: there

text: are

text: no

text: edges

text: between

text: C

text: and

text: C

text: and

text: by

text: the

text: deﬁnition

text: of

text: B

text: there

text: are

text: no

text: edges

text: between

text: C

text: and

text: A.

text: Let

text: us

text: now

text: show

text: that

text: vertices

text: v

text: C

text: have

text: many

text: neighbors

text: in

text: B

text: Claim

text: For

text: every

text: u

text: C

text: deg

text: (u)

text: Proof.

text: As

text: noted

text: above,

text: u

text: can

text: be

text: adjacent

text: only

text: to

text: the

text: vertices

text: in

text: B

text: D

text: Z.

text: First

text: observe

text: that

text: C

text: for

text: every

text: z

text: N

text: (u),

text: the

text: edge

text: u,

text: z

text: (G).

text: Indeed,

text: let

text: x

text: N

text: (u)

text: and

text: y

text: N

text: (x)

text: be

text: chosen

text: arbitrarily.

text: Since

text: y

text: C

text: and

text: z

text: D

text: there

text: exist

text: a

text: common

text: neighbor

text: of

text: y

text: and

text: z

text: which

text: encloses

text: a

text: C

text: Analogously,

text: we

text: show

text: u,

text: x

text: (G)

text: for

text: every

text: x

text: N

text: (u).

text: Consider

text: the

text: vertices

text: v

text: V

text: and

text: 5∈

text: w

text: D

text: with

text: u,

text: v

text: E(G)

text: and

text: v,

text: w

text: E(G)

text: witnessing

text: that

text: u

text: C

text: Since

text: x

text: B

text: and

text: w

text: D

text: the

text: two

text: vertices

text: must

text: have

text: a

text: common

text: neighbor

text: which

text: yields

text: u,

text: x

text: (G).

text: The

text: last

text: paragraph

text: shows

text: that

text: u√is

text: incident

text: to

text: at

text: most

text: Z

text: edges

text: that

text: do

text: not

text: occur

text: in

text: C

text: On

text: the

text: other

text: hand,

text: Z

text: D

text: So

text: if

text: deg

text: (v)

text: then

text: deg(v)

text: where

text: B

text: is

text: the

text: vertex

text: from

text: Claim

text: Therefore,

text: the

text: graph

text: obtained

text: by

text: removing

text: the

text: vertex

text: u

text: and

text: cloning

text: the

text: vertex

text: contradicts

text: Claim

text: Corollary

text: C

text: is

text: an

text: independent

text: set

text: in

text: G.

text: Proof.

text: Suppose

text: for

text: a

text: is

text: u,

text: with

text: u,

text: C

text: There

text: are

text: at

text: least

text: n

text: n

text: deg

text: (u)

text: A

text: edges

text: v,

text: w

text: with

text: v

text: N

text: (u)

text: and

text: w

text: N

text: (v).

text: However,

text: the

text: vertices

text: w

text: and

text: have

text: a

text: common

text: C

text: neighbor

text: in

text: B

text: v

text: and

text: hence

text: E(A

text: B

text: (G)

text: a

text: contradiction

text: with

text: Corollary

text: We

text: deﬁne

text: C

text: C

text: C

text: and

text: D

text: Z

text: (A

text: B

text: C

text: By

text: the

text: deﬁnition

text: of

text: the

text: sets

text: A,

text: B

text: and

text: 1∪

text: C

text: every

text: vertex

text: v

text: D

text: has

text: no

text: neigbors

text: in

text: A

text: B

text: We

text: now

text: concentrate

text: on

text: the

text: edges

text: between

text: D

text: and

text: C

text: Claim

text: For

text: every

text: v

text: D

text: deg

text: (v)

text: Proof.

text: The

text: vertex

text: v

text: can

text: be

text: adjacent

text: only

text: to

text: the

text: vertices

text: in

text: C

text: D

text: Z,

text: and

text: clearly

text: every

text: edge

text: v,

text: w

text: with

text: w

text: C

text: D

text: occurs

text: in

text: C

text: In

text: particular,

text: v

text: is

text: incident

text: to

text: at

text: most

text: edges

text: that

text: do

text: not

text: occur

text: in

text: C

text: As

text: in

text: Claim

text: if

text: deg

text: (v)

text: then

text: deg(v)

text: D

text: Z

text: Therefore,

text: removing

text: the

text: vertex

text: v

text: and

text: cloning

text: the

text: vertex

text: from

text: Claim

text: result

text: in

text: a

text: graph

text: contradicting

text: Claim

text: So

text: the

text: only

text: possible

text: edges

text: that

text: could

text: be

text: in

text: G

text: but

text: not

text: following

text: the

text: pattern

text: of

text: Construction

text: are

text: those

text: between

text: B

text: and

text: D

text: We

text: rule

text: them

text: out

text: in

text: the

text: following

text: claim.

text: Claim

text: E(B

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: | ∩ | | | N (u ) N (u ) > A /3. On the other hand, deg(u ) + deg(u ) < n/2. Therefore, replacing A 1 A 2 1 2 the vertices u and u with one new vertex adjacent to every vertex in D and a clone of the vertex 1 2 v from Claim 7.5 again yields a contradiction with Claim 7.2 | | ≥ | | ≥ We conclude that if C 2, then C = 2 and it contains vertices u , u and deg (u ) n/24. A A | | 1 2 D 1 ∈ Note that u or u is incident to at most A /2 edges that do not occur in C . Let u C be 1 2 ∈ ≥ 2k+1 A≥ this vertex and let u(cid:48) C be the other vertex. Since deg (u ) 2 and therefore deg (u(cid:48)) 2, · A − D 2 D there are at least 2 (deg (u ) 2) > deg (u) pairs of non-edges between N (u) and N (u(cid:48)). D 1 D { } ∈ D ∈D Therefore, removing the vertex u, adding all the edges w, w(cid:48) with w N (u) and w(cid:48) N (u(cid:48)), D D and adding a clone of the vertex v from Claim 7.5 contradicts Claim 7.2, which ﬁnishes the proof of the claim. | | | | It remains to show that we cannot have both C = 1 and C = 1. A B | | Claim 7.13. C = 1. ∈ ∈ Proof. Suppose for contradiction there are vertices u C and u C . Firstly, recall that ∩ ∅ a A b B N (u ) N (u ) = by Claim 7.11. D a D b | | | | Now let us prove that both N (u ) and N (u ) must have quite small sizes, say less than n/24. A a | B |b≥ Suppose, without loss of generality, that N (u ) n/24. Our aim now is to show that any edge A a ∈ incident to u is contained in some C . Consider any vertex v N (u ). Since deg (v) > | | − b 2k+1 ∈ B b A A 11εn, the vertices v and u have a common neighbor w A. Therefore, u wvu P gives a −a ∈ ∈a b \ { } copy of C , where P is a (2k 3)-vertex path in D between x N (u ) and x(cid:48) N (u ) x . 2k+1 C D a D b We conclude that every edge incident to u is in (G). Moreover, there is at least one edge in ∩ C b 2k+1 | | E(A, B) (G). But then consider a graph G(cid:48) obtained from G by removing at most B edges 2k+1 | | | | between u and B, and adding at least D > B missing edges between C and D. Since no edge b C from E(A, B) is in (G(cid:48)), the graph G(cid:48) contradicts Claim 7.2. 2k+1 | | | | It remains to consider the case when both N (u ) and N (u ) have sizes less than n/24. But A a B b then removing all the edges from, say, u to A, and adding all the missing edges between u and B a b yield a graph G(cid:48) that again contradicts Claim 7.2. (cid:0) (cid:1) This gives us a complete information on the structure of the extremal graph. ⊆ ∪· ∪· ∪· Corollary 7.14. E(G) E(A, B) E(B, C) E(C, D) D . 2 (cid:106) (cid:107) It follows that all the edges that do not occur in C are incident to vertices in B. 2k+1 | || | | | ≥ | | − | | | | Corollary 7.15. E(A, B) = A B and B A . Moreover, F (n) = n2 + 1 ( A + 1) B . 2k+1 4 ∈ G Finally, knowing the structure, it is straightforward to get the fact that G 2k+1 yields that | | (cid:98) − (cid:99) | | (cid:98) (cid:99) | | (cid:98) (cid:99) n A = (n 2)/6 , B = (n + 1)/6 and D = (2n + 1)/3 . (cid:4) (cid:5) 8 Concluding remarks For an n-vertex graph G with n2/4 +1 edges, we determined the asymptotic minimum number ≥ of the edges of G that occur in some copy of C in G, and for any k 3, the exact minimum number 5 of the edges that occur in C . Our results show that the pentagon case has a very diﬀerent 2k+1 behavior compared to all the longer odd cycles. These results conﬁrm a conjecture of Fu¨redi 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: (u

text: N

text: (u

text: A

text: On

text: the

text: other

text: hand,

text: deg(u

text: deg(u

text: Therefore,

text: replacing

text: A

text: A

text: the

text: vertices

text: u

text: and

text: u

text: with

text: one

text: new

text: vertex

text: adjacent

text: to

text: every

text: vertex

text: in

text: D

text: and

text: a

text: clone

text: of

text: the

text: vertex

text: v

text: from

text: Claim

text: again

text: yields

text: a

text: contradiction

text: with

text: Claim

text: We

text: conclude

text: that

text: if

text: C

text: then

text: C

text: and

text: it

text: contains

text: vertices

text: u

text: u

text: and

text: deg

text: (u

text: A

text: A

text: D

text: Note

text: that

text: u

text: or

text: u

text: is

text: incident

text: to

text: at

text: most

text: A

text: edges

text: that

text: do

text: not

text: occur

text: in

text: C

text: Let

text: u

text: C

text: be

text: A≥

text: this

text: vertex

text: and

text: let

text: C

text: be

text: the

text: other

text: vertex.

text: Since

text: deg

text: (u

text: and

text: therefore

text: deg

text: A

text: D

text: D

text: there

text: are

text: at

text: least

text: (deg

text: (u

text: deg

text: (u)

text: pairs

text: of

text: non-edges

text: between

text: N

text: (u)

text: and

text: N

text: D

text: D

text: D

text: ∈D

text: Therefore,

text: removing

text: the

text: vertex

text: u,

text: adding

text: all

text: the

text: edges

text: w,

text: with

text: w

text: N

text: (u)

text: and

text: N

text: D

text: D

text: and

text: adding

text: a

text: clone

text: of

text: the

text: vertex

text: v

text: from

text: Claim

text: contradicts

text: Claim

text: which

text: ﬁnishes

text: the

text: proof

text: of

text: the

text: claim.

text: It

text: remains

text: to

text: show

text: that

text: we

text: cannot

text: have

text: both

text: C

text: and

text: C

text: A

text: B

text: Claim

text: C

text: Proof.

text: Suppose

text: for

text: contradiction

text: there

text: are

text: vertices

text: u

text: C

text: and

text: u

text: C

text: Firstly,

text: recall

text: that

text: a

text: A

text: b

text: B

text: N

text: (u

text: N

text: (u

text: by

text: Claim

text: D

text: a

text: D

text: b

text: Now

text: let

text: us

text: prove

text: that

text: both

text: N

text: (u

text: and

text: N

text: (u

text: must

text: have

text: quite

text: small

text: sizes,

text: say

text: less

text: than

text: A

text: a

text: B

text: |b≥

text: Suppose,

text: without

text: loss

text: of

text: generality,

text: that

text: N

text: (u

text: Our

text: aim

text: now

text: is

text: to

text: show

text: that

text: any

text: edge

text: A

text: a

text: incident

text: to

text: u

text: is

text: contained

text: in

text: some

text: C

text: Consider

text: any

text: vertex

text: v

text: N

text: (u

text: Since

text: deg

text: (v)

text: b

text: B

text: b

text: A

text: A

text: the

text: vertices

text: v

text: and

text: u

text: have

text: a

text: common

text: neighbor

text: w

text: A.

text: Therefore,

text: u

text: wvu

text: P

text: gives

text: a

text: −a

text: ∈a

text: b

text: copy

text: of

text: C

text: where

text: P

text: is

text: a

text: path

text: in

text: D

text: between

text: x

text: N

text: (u

text: and

text: N

text: (u

text: x

text: C

text: D

text: a

text: D

text: b

text: We

text: conclude

text: that

text: every

text: edge

text: incident

text: to

text: u

text: is

text: in

text: (G).

text: Moreover,

text: there

text: is

text: at

text: least

text: one

text: edge

text: in

text: C

text: b

text: E(A,

text: B)

text: (G).

text: But

text: then

text: consider

text: a

text: graph

text: obtained

text: from

text: G

text: by

text: removing

text: at

text: most

text: B

text: edges

text: between

text: u

text: and

text: B,

text: and

text: adding

text: at

text: least

text: D

text: B

text: missing

text: edges

text: between

text: C

text: and

text: D.

text: Since

text: no

text: edge

text: b

text: C

text: from

text: E(A,

text: B)

text: is

text: in

text: the

text: graph

text: contradicts

text: Claim

text: It

text: remains

text: to

text: consider

text: the

text: case

text: when

text: both

text: N

text: (u

text: and

text: N

text: (u

text: have

text: sizes

text: less

text: than

text: But

text: A

text: a

text: B

text: b

text: then

text: removing

text: all

text: the

text: edges

text: from,

text: say,

text: u

text: to

text: A,

text: and

text: adding

text: all

text: the

text: missing

text: edges

text: between

text: u

text: and

text: B

text: a

text: b

text: yield

text: a

text: graph

text: that

text: again

text: contradicts

text: Claim

text: This

text: gives

text: us

text: a

text: complete

text: information

text: on

text: the

text: structure

text: of

text: the

text: extremal

text: graph.

text: Corollary

text: E(G)

text: E(A,

text: B)

text: E(B,

text: C)

text: E(C,

text: D)

text: D

text: It

text: follows

text: that

text: all

text: the

text: edges

text: that

text: do

text: not

text: occur

text: in

text: C

text: are

text: incident

text: to

text: vertices

text: in

text: B.

text: Corollary

text: E(A,

text: B)

text: A

text: B

text: and

text: B

text: A

text: Moreover,

text: F

text: (n)

text: A

text: B

text: G

text: Finally,

text: knowing

text: the

text: structure,

text: it

text: is

text: straightforward

text: to

text: get

text: the

text: fact

text: that

text: G

text: yields

text: that

text: n

text: A

text: (n

text: B

text: (n

text: and

text: D

text: Concluding

text: remarks

text: For

text: an

text: n-vertex

text: graph

text: G

text: with

text: edges,

text: we

text: determined

text: the

text: asymptotic

text: minimum

text: number

text: of

text: the

text: edges

text: of

text: G

text: that

text: occur

text: in

text: some

text: copy

text: of

text: C

text: in

text: G,

text: and

text: for

text: any

text: k

text: the

text: exact

text: minimum

text: number

text: of

text: the

text: edges

text: that

text: occur

text: in

text: C

text: Our

text: results

text: show

text: that

text: the

text: pentagon

text: case

text: has

text: a

text: very

text: diﬀerent

text: behavior

text: compared

text: to

text: all

text: the

text: longer

text: odd

text: cycles.

text: These

text: results

text: conﬁrm

text: a

text: conjecture

text: of

text: Fu¨redi

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: and Maleki, who proved the optimal asymptotic bounds under a stronger assumption that G has (1/4 + ε)n2 edges. (cid:4) (cid:5) Our main tool was an application of techniques from ﬁnite forcibility in the setting of ﬂag algebras, combined with stability results on triangle-free graphs. This was crucial for dealing with n-vertex graphs that have only n2/4 + 1 edges. We believe that our approach can be adapted to various other scenarios, and we intend to investigate this direction further. We were also able to guide ﬂag algebras to give us additional structural information for extremal conﬁgurations which yielded the corresponding stability results. These stability results allowed us to fully describe the structure of all the suﬃciently large tight constructions. (cid:4) (cid:5) If G contains αn2 edges for some α > 1/4, then a standard averaging argument yields that G must contain much more edges that occur in C , for k being ﬁxed, than Theorems 1.3 and 1.4 2k+1 guarantee for n2/4 + 1 edges. However, the averaging argument yields only a weak improvement. In [19], Fu¨redi and Maleki determined an asymptotically optimal lower bound for this problem. Note that the corresponding approximate result for triangles was proven by Fu¨redi and Maleki in [18]. Fu¨redi and Maleki [19] also considered a more general question, where instead of minimizing the number of edges that occur in odd cycles of a ﬁxed length, one minimizes the number of edges that occur in copies of F for some ﬁxed graph F . If the graph F has the chromatic number χ = 3, they obtained an asymptotically tight solution to this question. However, for graphs F with the ≥ chromatic number χ 4, these questions are widely open. (cid:4) (cid:5) All the problems we mentioned so far were concerned with the number of edges that occur in copies of some ﬁxed graph F . In [13], Erd˝os, Faudree and Rousseau also determined what is the minimum number of vertices that occur in triangles for n-vertex graphs with n2/4 + 1 edges. For (cid:4) (cid:5) odd cycles of longer length, they obtained asymptotically tight lower bound. The second author together with Shagnik Das, Tibor Szab´o and Tuan Tran [9] recently obtained tight lower bounds also for graphs that have at least n2/4 + 1 edges. Acknowledgments. The authors thank Zoltan Fu¨redi and Zeinab Maleki for discussing the results of [19] and the relation to the results obtained in this paper, and to Shoham Letzter for her suggestions regarding the results presented in Sections and . We also want to thank Jake Cooper and Dan Kr´al’ for fruitful discussions at the beginning of this project. All of these discussions greatly improved the presentation of our results. References [1] T. Austin and T. Tao. Testability and repair of hereditary hypergraph properties. Random Structures Algorithms, 36(4):373–463, July 2010. [2] R. Baber. Tur´an densities of hypercubes. Submitted, 2012. [3] R. Baber and J. Talbot. Hypergraphs do jump. Combin. Probab. Comput., 20(2):161–171, 2011. [4] R. Baber and J. Talbot. A Solution to the 2/3 Conjecture. SIAM J. Discrete Math., 28(2):756– 766, 2014. [5] J. Balogh, P. Hu, B. Lidicky´, and H. Liu. Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. European J. Combin., 35:75–85, 2014. 31

text: and

text: Maleki,

text: who

text: proved

text: the

text: optimal

text: asymptotic

text: bounds

text: under

text: a

text: stronger

text: assumption

text: that

text: G

text: has

text: edges.

text: Our

text: main

text: tool

text: was

text: an

text: application

text: of

text: techniques

text: from

text: ﬁnite

text: forcibility

text: in

text: the

text: setting

text: of

text: ﬂag

text: algebras,

text: combined

text: with

text: stability

text: results

text: on

text: triangle-free

text: graphs.

text: This

text: was

text: crucial

text: for

text: dealing

text: with

text: n-vertex

text: graphs

text: that

text: have

text: only

text: edges.

text: We

text: believe

text: that

text: our

text: approach

text: can

text: be

text: adapted

text: to

text: various

text: other

text: scenarios,

text: and

text: we

text: intend

text: to

text: investigate

text: this

text: direction

text: further.

text: We

text: were

text: also

text: able

text: to

text: guide

text: ﬂag

text: algebras

text: to

text: give

text: us

text: additional

text: structural

text: information

text: for

text: extremal

text: conﬁgurations

text: which

text: yielded

text: the

text: corresponding

text: stability

text: results.

text: These

text: stability

text: results

text: allowed

text: us

text: to

text: fully

text: describe

text: the

text: structure

text: of

text: all

text: the

text: suﬃciently

text: large

text: tight

text: constructions.

text: If

text: G

text: contains

text: edges

text: for

text: some

text: α

text: then

text: a

text: standard

text: averaging

text: argument

text: yields

text: that

text: G

text: must

text: contain

text: much

text: more

text: edges

text: that

text: occur

text: in

text: C

text: for

text: k

text: being

text: ﬁxed,

text: than

text: Theorems

text: and

text: guarantee

text: for

text: edges.

text: However,

text: the

text: averaging

text: argument

text: yields

text: only

text: a

text: weak

text: improvement.

text: In

text: Fu¨redi

text: and

text: Maleki

text: determined

text: an

text: asymptotically

text: optimal

text: lower

text: bound

text: for

text: this

text: problem.

text: Note

text: that

text: the

text: corresponding

text: approximate

text: result

text: for

text: triangles

text: was

text: proven

text: by

text: Fu¨redi

text: and

text: Maleki

text: in

text: Fu¨redi

text: and

text: Maleki

text: also

text: considered

text: a

text: more

text: general

text: question,

text: where

text: instead

text: of

text: minimizing

text: the

text: number

text: of

text: edges

text: that

text: occur

text: in

text: odd

text: cycles

text: of

text: a

text: ﬁxed

text: length,

text: one

text: minimizes

text: the

text: number

text: of

text: edges

text: that

text: occur

text: in

text: copies

text: of

text: F

text: for

text: some

text: ﬁxed

text: graph

text: F

text: If

text: the

text: graph

text: F

text: has

text: the

text: chromatic

text: number

text: χ

text: they

text: obtained

text: an

text: asymptotically

text: tight

text: solution

text: to

text: this

text: question.

text: However,

text: for

text: graphs

text: F

text: with

text: the

text: chromatic

text: number

text: χ

text: these

text: questions

text: are

text: widely

text: open.

text: All

text: the

text: problems

text: we

text: mentioned

text: so

text: far

text: were

text: concerned

text: with

text: the

text: number

text: of

text: edges

text: that

text: occur

text: in

text: copies

text: of

text: some

text: ﬁxed

text: graph

text: F

text: In

text: Erd˝os,

text: Faudree

text: and

text: Rousseau

text: also

text: determined

text: what

text: is

text: the

text: minimum

text: number

text: of

text: vertices

text: that

text: occur

text: in

text: triangles

text: for

text: n-vertex

text: graphs

text: with

text: edges.

text: For

text: odd

text: cycles

text: of

text: longer

text: length,

text: they

text: obtained

text: asymptotically

text: tight

text: lower

text: bound.

text: The

text: second

text: author

text: together

text: with

text: Shagnik

text: Das,

text: Tibor

text: Szab´o

text: and

text: Tuan

text: Tran

text: recently

text: obtained

text: tight

text: lower

text: bounds

text: also

text: for

text: graphs

text: that

text: have

text: at

text: least

text: edges.

text: Acknowledgments.

text: The

text: authors

text: thank

text: Zoltan

text: Fu¨redi

text: and

text: Zeinab

text: Maleki

text: for

text: discussing

text: the

text: results

text: of

text: and

text: the

text: relation

text: to

text: the

text: results

text: obtained

text: in

text: this

text: paper,

text: and

text: to

text: Shoham

text: Letzter

text: for

text: her

text: suggestions

text: regarding

text: the

text: results

text: presented

text: in

text: Sections

text: and

text: We

text: also

text: want

text: to

text: thank

text: Jake

text: Cooper

text: and

text: Dan

text: Kr´al’

text: for

text: fruitful

text: discussions

text: at

text: the

text: beginning

text: of

text: this

text: project.

text: All

text: of

text: these

text: discussions

text: greatly

text: improved

text: the

text: presentation

text: of

text: our

text: results.

text: References

text: T.

text: Austin

text: and

text: T.

text: Tao.

text: Testability

text: and

text: repair

text: of

text: hereditary

text: hypergraph

text: properties.

text: Random

text: Structures

text: Algorithms,

text: July

text: R.

text: Baber.

text: Tur´an

text: densities

text: of

text: hypercubes.

text: Submitted,

text: R.

text: Baber

text: and

text: J.

text: Talbot.

text: Hypergraphs

text: do

text: jump.

text: Combin.

text: Probab.

text: Comput.,

text: R.

text: Baber

text: and

text: J.

text: Talbot.

text: A

text: Solution

text: to

text: the

text: Conjecture.

text: SIAM

text: J.

text: Discrete

text: Math.,

text: J.

text: Balogh,

text: P.

text: Hu,

text: B.

text: Lidicky´,

text: and

text: H.

text: Liu.

text: Upper

text: bounds

text: on

text: the

text: size

text: of

text: and

text: subgraphs

text: of

text: the

text: hypercube.

text: European

text: J.

text: Combin.,


text: red/blue-colored graphs H with a ﬁxed embedding of σ, i.e., an injective mapping θ from V (σ) H to V (H) such that θ is an isomorphism between σ and H[im(θ)]. The elements of σ are called ∈ N σ-ﬂags and the subgraph induced by im(θ) is called the root of a σ-ﬂag. For every (cid:96) , we deﬁne H ⊂ H H σ σ to be the set of all (cid:96)-vertex σ-ﬂags from σ. (cid:96) ∈ H ∈ H For two σ-ﬂags H σ and H(cid:48) σ with the embeddings of σ given by θ and θ(cid:48), respectively, − we set p(H, H(cid:48)) to be the probability that a randomly chosen subset of v(H) v(σ) vertices in \ V (H(cid:48)) θ(cid:48)(V (σ)) together with θ(cid:48)(V (σ)) induces a σ-ﬂag that is isomorphic to H through an isomorphism f that preserves the embedding of σ. In other words, the isomorphism f has to RH H satisfy f(θ(cid:48)) = θ. Let σ be the set of all formal linear combinations of elements of σ with real K (cid:88) RH coeﬃcients, and let σ be the linear subspace of σ generated by all the linear combinations of the form − · H p(H, H(cid:48)) H(cid:48). (cid:48) H ∈Hσ v(H)+1 A RH K A We deﬁne σ to be σ factored by σ and, analogously to the case for the algebra , we let the K A element corresponding to σ to be the zero element of σ. H We now deﬁne the product of two elements from σ. Fix two integers (cid:96) and (cid:96) and let − ∈ H ∈ H ∈ H 1 2 (cid:96) := (cid:96) + (cid:96) v(σ). Let H σ , H σ and H σ be σ-ﬂags, and let θ be the ﬁxed 1 2 1 (cid:96)1 2 (cid:96)2 (cid:96) A embedding of σ in H. Similarly to the deﬁnition of the multiplication for , we deﬁne p(H , H ; H) \ − 1 2 to be the probability that a randomly chosen subset of V (H) θ(V (σ)) of size (cid:96) v(σ) and its \ − 1 complement in V (H) θ(V (σ)) of size (cid:96) v(σ), extend θ(V (σ)) in H to σ-ﬂags isomorphic to H 2 (cid:88) 1 and H , respectively. We set 2 × · H H := p(H , H ; H) H. 1 2 1 2 H∈Hσ − v(H1)+v(H2) v(σ) A H As in the case for the algebra , the deﬁnition of the product for the elements of σ naturally A K extends to σ, and the unique σ-ﬂag of size v(σ), modulo σ, represents the neutral element of A the product in σ. A A We have introduced the ﬂag algebras and σ on red/blue-colored graphs. It is easy to see A A F that the same exposition can be used to deﬁne ﬂag algebras and σ on -free red/blue-colored F F F H H graphs, where is a ﬁxed ﬁnite family of red/blue-colored graphs, simply by replacing and σ F F with the set of all red/blue-colored -free graphs and all -free σ-ﬂags, respectively. F Consider an inﬁnite sequence (G ) N of red/blue-colored almost -free graphs with increasing i i∈ orders. We say that the sequence (G ) N is convergent if the probabilities p(H, G ) converge for ∈ H i i∈ i every H . It follows that every inﬁnite sequence (G ) N has a convergent subsequence. Fix a iFi∈ convergent sequence (G ) N of red/blue-colored almost -free graphs with increasing orders. For ∈ H i i∈ A every H , we set φ(H) = lim p(H, G ), and then linearly extend φ to the elements of . i→∞ i We usually refer to the mapping φ as to the limit of the sequence. The obtained mapping φ is an A R ∈ H algebra homomorphism from to , see [33, Theorem 3.3a]. Moreover, for every H , it holds ≥ that φ(H) 0. A R Let Hom+( , ) be the set of all such homomorphisms, i.e., the set of all homomorphisms ψ A R ≥ ∈ H from the algebra to such that ψ(H) 0 for every H . It is interesting to see that this set is exactly the set of all the limits of convergent sequences [33, Theorem 3.3b]. Since the convergent F ∈ F sequence (G ) N is almost -free, it follows that φ(F ) = 0 for any F . Therefore, the algebra i i∈ A → R F homomorphism φ : is supported only on -free graphs, and hence it can be viewed also as 7

text: red/blue-colored

text: graphs

text: H

text: with

text: a

text: ﬁxed

text: embedding

text: of

text: σ,

text: i.e.,

text: an

text: injective

text: mapping

text: θ

text: from

text: V

text: (σ)

text: H

text: to

text: V

text: (H)

text: such

text: that

text: θ

text: is

text: an

text: isomorphism

text: between

text: σ

text: and

text: H[im(θ)].

text: The

text: elements

text: of

text: σ

text: are

text: called

text: N

text: σ-ﬂags

text: and

text: the

text: subgraph

text: induced

text: by

text: im(θ)

text: is

text: called

text: the

text: root

text: of

text: a

text: σ-ﬂag.

text: For

text: every

text: we

text: deﬁne

text: H

text: H

text: H

text: σ

text: σ

text: to

text: be

text: the

text: set

text: of

text: all

text: σ-ﬂags

text: from

text: σ.

text: H

text: H

text: For

text: two

text: σ-ﬂags

text: H

text: σ

text: and

text: σ

text: with

text: the

text: embeddings

text: of

text: σ

text: given

text: by

text: θ

text: and

text: respectively,

text: we

text: set

text: p(H,

text: to

text: be

text: the

text: probability

text: that

text: a

text: randomly

text: chosen

text: subset

text: of

text: v(H)

text: v(σ)

text: vertices

text: in

text: V

text: (σ))

text: together

text: with

text: (σ))

text: induces

text: a

text: σ-ﬂag

text: that

text: is

text: isomorphic

text: to

text: H

text: through

text: an

text: isomorphism

text: f

text: that

text: preserves

text: the

text: embedding

text: of

text: σ.

text: In

text: other

text: words,

text: the

text: isomorphism

text: f

text: has

text: to

text: RH

text: H

text: satisfy

text: θ.

text: Let

text: σ

text: be

text: the

text: set

text: of

text: all

text: formal

text: linear

text: combinations

text: of

text: elements

text: of

text: σ

text: with

text: real

text: K

text: RH

text: coeﬃcients,

text: and

text: let

text: σ

text: be

text: the

text: linear

text: subspace

text: of

text: σ

text: generated

text: by

text: all

text: the

text: linear

text: combinations

text: of

text: the

text: form

text: H

text: p(H,

text: H

text: ∈Hσ

text: A

text: RH

text: K

text: A

text: We

text: deﬁne

text: σ

text: to

text: be

text: σ

text: factored

text: by

text: σ

text: and,

text: analogously

text: to

text: the

text: case

text: for

text: the

text: algebra

text: we

text: let

text: the

text: K

text: A

text: element

text: corresponding

text: to

text: σ

text: to

text: be

text: the

text: zero

text: element

text: of

text: σ.

text: H

text: We

text: now

text: deﬁne

text: the

text: product

text: of

text: two

text: elements

text: from

text: σ.

text: Fix

text: two

text: integers

text: and

text: and

text: let

text: H

text: H

text: H

text: v(σ).

text: Let

text: H

text: σ

text: H

text: σ

text: and

text: H

text: σ

text: be

text: σ-ﬂags,

text: and

text: let

text: θ

text: be

text: the

text: ﬁxed

text: A

text: embedding

text: of

text: σ

text: in

text: H.

text: Similarly

text: to

text: the

text: deﬁnition

text: of

text: the

text: multiplication

text: for

text: we

text: deﬁne

text: p(H

text: H

text: H)

text: to

text: be

text: the

text: probability

text: that

text: a

text: randomly

text: chosen

text: subset

text: of

text: V

text: (H)

text: θ(V

text: (σ))

text: of

text: size

text: v(σ)

text: and

text: its

text: complement

text: in

text: V

text: (H)

text: θ(V

text: (σ))

text: of

text: size

text: v(σ),

text: extend

text: θ(V

text: (σ))

text: in

text: H

text: to

text: σ-ﬂags

text: isomorphic

text: to

text: H

text: and

text: H

text: respectively.

text: We

text: set

text: H

text: H

text: p(H

text: H

text: H)

text: H.

text: H∈Hσ

text: v(σ)

text: A

text: H

text: As

text: in

text: the

text: case

text: for

text: the

text: algebra

text: the

text: deﬁnition

text: of

text: the

text: product

text: for

text: the

text: elements

text: of

text: σ

text: naturally

text: A

text: K

text: extends

text: to

text: σ,

text: and

text: the

text: unique

text: σ-ﬂag

text: of

text: size

text: v(σ),

text: modulo

text: σ,

text: represents

text: the

text: neutral

text: element

text: of

text: A

text: the

text: product

text: in

text: σ.

text: A

text: A

text: We

text: have

text: introduced

text: the

text: ﬂag

text: algebras

text: and

text: σ

text: on

text: red/blue-colored

text: graphs.

text: It

text: is

text: easy

text: to

text: see

text: A

text: A

text: F

text: that

text: the

text: same

text: exposition

text: can

text: be

text: used

text: to

text: deﬁne

text: ﬂag

text: algebras

text: and

text: σ

text: on

text: -free

text: red/blue-colored

text: F

text: F

text: F

text: H

text: H

text: graphs,

text: where

text: is

text: a

text: ﬁxed

text: ﬁnite

text: family

text: of

text: red/blue-colored

text: graphs,

text: simply

text: by

text: replacing

text: and

text: σ

text: F

text: F

text: with

text: the

text: set

text: of

text: all

text: red/blue-colored

text: -free

text: graphs

text: and

text: all

text: -free

text: σ-ﬂags,

text: respectively.

text: F

text: Consider

text: an

text: inﬁnite

text: sequence

text: (G

text: N

text: of

text: red/blue-colored

text: almost

text: -free

text: graphs

text: with

text: increasing

text: i

text: i∈

text: orders.

text: We

text: say

text: that

text: the

text: sequence

text: (G

text: N

text: is

text: convergent

text: if

text: the

text: probabilities

text: p(H,

text: G

text: converge

text: for

text: H

text: i

text: i∈

text: i

text: every

text: H

text: It

text: follows

text: that

text: every

text: inﬁnite

text: sequence

text: (G

text: N

text: has

text: a

text: convergent

text: subsequence.

text: Fix

text: a

text: iFi∈

text: convergent

text: sequence

text: (G

text: N

text: of

text: red/blue-colored

text: almost

text: -free

text: graphs

text: with

text: increasing

text: orders.

text: For

text: H

text: i

text: i∈

text: A

text: every

text: H

text: we

text: set

text: φ(H)

text: lim

text: p(H,

text: G

text: and

text: then

text: linearly

text: extend

text: φ

text: to

text: the

text: elements

text: of

text: i→∞

text: i

text: We

text: usually

text: refer

text: to

text: the

text: mapping

text: φ

text: as

text: to

text: the

text: limit

text: of

text: the

text: sequence.

text: The

text: obtained

text: mapping

text: φ

text: is

text: an

text: A

text: R

text: H

text: algebra

text: homomorphism

text: from

text: to

text: see

text: Theorem

text: Moreover,

text: for

text: every

text: H

text: it

text: holds

text: that

text: φ(H)

text: A

text: R

text: Let

text: Hom+(

text: be

text: the

text: set

text: of

text: all

text: such

text: homomorphisms,

text: i.e.,

text: the

text: set

text: of

text: all

text: homomorphisms

text: ψ

text: A

text: R

text: H

text: from

text: the

text: algebra

text: to

text: such

text: that

text: ψ(H)

text: for

text: every

text: H

text: It

text: is

text: interesting

text: to

text: see

text: that

text: this

text: set

text: is

text: exactly

text: the

text: set

text: of

text: all

text: the

text: limits

text: of

text: convergent

text: sequences

text: Theorem

text: Since

text: the

text: convergent

text: F

text: F

text: sequence

text: (G

text: N

text: is

text: almost

text: -free,

text: it

text: follows

text: that

text: φ(F

text: for

text: any

text: F

text: Therefore,

text: the

text: algebra

text: i

text: i∈

text: A

text: R

text: F

text: homomorphism

text: φ

text: is

text: supported

text: only

text: on

text: -free

text: graphs,

text: and

text: hence

text: it

text: can

text: be

text: viewed

text: also

text: as