Non-residues and primitive roots in Beatty sequences

Abstract

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p1/2+ε ≤ N ≤ p, then among the first N elements of Bα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) emits a shaper estimate.