Short character sums with beatty sequences

We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence {⌊αn + β⌋: n = 1,2,...}, where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for fixed α,β and a small real number ε > 0, if p is sufficiently large and p ^1/3+ε ≤ N ≤ p ^1/3+ε, then among the first N elements of the Beatty sequence there are N/2 + o(N) quadratic non-residues modulo p. If additional information about the Diophantine properties of α is available, then the error term o(N) admits a sharper estimate. This work improves and extends several previous results due to M. Z. Garaev and S. N. Preobrazhenskiǐ.