On the maxima of Littlewood polynomials on $[-1,1]$

A Littlewood polynomial is a polynomial of the form $$f_n(x)=\sum _{k=0}^n{\epsilon }_k{x}^k$$ with ${\epsilon }_k\in \{-1,1\}$. Let $({\epsilon }_k{)}_{k\ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of ${\max }_{x\in [-1,1]}\mathrm{\mid }{f}_n(x)\mathrm{\mid }$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log (\underset{x\in [-1,1]}{\max }\mathrm{\mid }{f}_n(x)\mathrm{\mid }\mathrm{/}\sqrt{n})}{(\log \log n{)}^{1\mathrm{/}3}}=-(\frac{3{\pi }^2}{4}{)}^{1\mathrm{/}3}\mathrm{.}$$