TITLE: Minimal period of arithmetic progressions occurring in sets of positive density.
QUESTION [6 upvotes]: Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ arithmetic progression
$$
\{a, a+r, a+2r, \dots, a+(k-1)r\}.
$$
Thus, the finiteness of $R_k$ is Szemeredi's theorem.  
What, if anything, is known about how $R_k$ grows?  More precisely, the question I am most interested in, without any luck so far, is the following: 

Does there exist an example of a set $A$ for which $R_k$ grows with $k$, with, if possible, a lower bound on $R_k$?  This lower bound may depend on $A$, but I am hoping for an explicit dependence on $k$.

If there are results in the other direction -- upper bounds on $R_k$ -- I would be interested to hear about those as well.  Such a result could be considered a quantitative strengthening of Szemeredi's theorem, so perhaps this is asking for a lot.

REPLY [2 votes]: It seems that a nice example is the set $A$ of positive integers which have an even number of 1's in their binary expansion, although I don't see a reasonable lower bound on $R_k(A)$ for now. A quick computation suggests that $R_2(A) = \dots = R_8(A) = 3$, $R_9(A) = R_{10}(A) = 9$, $R_{11}(A) = \dots = R_{20}(A) = 15$, $R_{21}(A) = \dots = R_{32}(A) = 31$, $R_{33}(A) = R_{34}(A) = 33$ and $R_{35}(A) = \dots = R_{68}(A) = 63$.