TITLE: L1 distance from a trigonometric susbspace
QUESTION [5 upvotes]: How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in \mathbb{Z}\setminus F \right)$ is less or equal than $1$? 
This problem I have found oryginally stated as follows:
For which subset $F\subset\mathbb{Z}$ there exists a function $f\in L^{1}$ such that $\widehat{f}(k)=1$ for $k\in F$ and $\| f \|_1=1$.
I have known some estimates of $L^{1} $ norm of $S_F$, however, such estimates gives an error which leads to many cases left.

REPLY [2 votes]: I'm not sure about your version of the problem, but the following result of Helson from 1955 partially answers the original problem:
Claim: Let $\mu$ be a measure on $\mathbb{T}$ such that $\hat{\mu}(n)=0$ or $1$ for $n=0,\pm1,\pm2,\ldots.$  Then $\{n\in\mathbb{N}\colon\hat{\mu}(n)=1\}=(F_{1}\cup V)\setminus F_{2}$ with $F_{1},F_{2}$ finite sets and $V$ a periodic set, i.e. a finite sum of arithmetic progressions.
Proof:  We argue by contradiction.  Suppose the conclusion is false.  Then without loss of generality, we can find two increasing sequences $n(t),m(t)$ such that $n(t),m(t)\to\infty$ as $t\to\infty$, $\hat{\mu}(n(t)-j)=\hat{\mu}(m(t)-j)$ for $j=1,\ldots,t-1$, but $\hat{\mu}(n(t))\neq\hat{\mu}(m(t))$.  Write $\mu=fdm+\mu_{s}$ with $dm$ Lebesgue measure and $\mu_{s}$ singular.  By our assumptions (ignoring trivial cases, e.g. when $\mu$ is a finite trigonometric polynomial) $\mu_{s}\neq0$.  (If $\mu_{s}=0$, the Fourier coefficients would vanish at infinity by Riemann-Lebesgue.)  Now, consider the measure
$$\nu_{t}=(e^{-im(t)}-e^{-in(t)})\mu_{s}$$
As $t\to\infty$, we get a weak$^{*}$ limit $\nu\in L_{1}(|\mu_{s}|)$ by Banach-Alaoglu.  Note that $\nu$ is a singular measure, since it is the weak$^{*}$ limit of singular measures.  However, $\hat{\nu}(n)=0$ for $n<0$, so $\nu$ is absolutely continuous, by the F.-M. Riesz Theorem.  This contradiction gives our desired result.