A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$.

I'm interested in a "good" lower bound on the $L^1$-norm of Littlewood polynomials on the unit circle in the complex plane.(which I denote by $\|\cdot \|_1$) To be more precise, consider the polynomial $p(z) = 1+z+\cdots+z^n$, what can be said about $$a_n = \frac{\min_{q\in \cal{L}_n} \| q\|_1}{\| p\|_1}.$$ Does $a_n \to 1$ as $n\to \infty$?

1more comment