Type: Article

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

W. D. Banks; M. Z. Garaev; D. R. Heath-Brown; I. E. Shparlinski;

Abstract:

We show that for any fixed epsilon > 0, there axe numbers delta > 0 and p(0) >=, 2 with the following property: for every prime p >= p(0) and every integer N such that p(1/(4 root e)+epsilon) <= N <= p, the sequence 1, 2,..., N contains at least delta 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