Density of non-residues in Burgess-type intervals and applications

Abstract

We show that for any fixed ε > 0, there are numbers δ > 0 and p0 ⩾ 2 with the following property: for every prime p ⩾ p0 and every integer N such that p1/(4√e)+ε ⩽ N ⩽ p, the sequence 1, 2, …, N contains at least δ N quadratic non-residues modulo p. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski-Shapiro sequences.