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

Let T(x) = Sigma(j=0)(n-1) +/- e(ijx) where +/- stands for a random choice of sign with equal probability. The first author recently showed that for a ny epsilon > 0 and most choices of sign, min(x is an element of 2[0,2 pi)) \T(x)\ < n(-1/2+epsilon), provided n is large. In this paper we show that t he power n(-1/2) is optimal. More precisely, for sufficiently small epsilon > 0 and large n most choices of sign satisfy min(x is an element of[0, 2 p i)) \T(x)\ > epsilon n(-1/2). Furthermore, we study the case of more genera l random coefficients and applications of our methods to complex zeros of r andom polynomials.