On minima of the absolute value of certain random exponential sums

Let T_n(x) = Σⁿ_j=1 ±e^2πij2x where ± stands for a random choice of sign with equal probability. It is shown here that with high probability min_xε[0,1) |T_n(x)| < n^-σ provided n is large and σ < 1/12. Similar results are proved for other powers than squares. The problem of determining the optimal σ is open. For the case T_n(x)= Σⁿ_j=1r_je^2πijdx, where d = 2, 3,... is fixed and with standard normal r_j we show that the minima are typically on the order of n^-d+½ with high probability and for large n.