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Let T (x )= P n 1 j=0 e ijx where stands for a random choice of sign with equal probability. The rst author recently showed that for any >0 and most choices of sign, minx2(0;2) jT(x)j <n 1 =2+, provided n is large. In this paper we show that the power n 1=2 is optimal. More pre- cisely, for suciently small > 0a nd large nmost choices of sign satisfy minx2(0;2) jT(x)j >n 1 = 2. Furthermore, we study the case of more gen- eral random coecients and applications of our methods to complex zeros of random polynomials.

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