On Minima of the Absolute Value of Certain Random Exponential Sums

Let Tn(x) = Pn j=1 ±e2πij 2x where ± stands for a random choice of sign with equal probability. It is shown here that with high probability minx∈[0,1) |Tn(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 Tn(x) = Pn j=1 rje 2πijdx, where d = 2, 3, . . . is fixed and with standard normal rj we show that the minima are typically on the order of n−d+ 1 2 with high probability and for large n.