Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle

Let $T(x)=\sum _{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min _{x\in[0,2\pi)}|T(x)|<n^{-1/2+\epsilon}$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon>0$ and large $n$ most choices of sign satisfy $\min _{x\in[0,2\pi)}|T(x)|> \epsilon n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.