Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients. He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether there exists a sequence $f_n$ in $\mathscr F$ such that $$A\leq\frac{|f_n(\theta)|}{\sqrt{n+1}}\leq B$$ for large n's.

**1.** I'd like to know if this is still open and would be grateful if someone could provide references for recent survey papers.

**2.** If the conjecture is not settled yet, I was thinking if one can apply integer programming methods to tackle the problem?

**Update 1** As pointed out by Fedor, J-P. Kahane's work gives an elegant proof for polynomials with unimodular coefficients. His result is as follows(I didn't translate it to English as seems rather easy to understand ):

Theoreme.Il existe une suite de polynomes $$P_n(z)=\sum_{m=1}^{n}a_{m,n}z^m,\,(|a_{m,n}| = 1; \, n=1,2,\ldots,m=1,\ldots,n)$$ et une suite $\epsilon_n$ positive tendant vers 0 telles que pour tout z de module 1 on ait$$(1-\epsilon_n)\sqrt{n}\leq|P_n(z)|\leq(1+\epsilon_n)\sqrt{n}.$$

Numerical experiments suggest there might be hope to construct a sequence of $P_n\in\mathbb {Z}_2[x]$ which satisfies Kahane's asymptotics.

[L]*Littlewood, J. E.*, **On polynomials $\sum^n\pm z^m, \sum^n e^{\alpha_mi}z^m, z = e^{\theta i}$**, J. Lond. Math. Soc. 41, 367-376 (1966). ZBL0142.32603.