Short character sums with beatty sequences

William D. Banks*, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)539-547
Number of pages9
JournalMathematical Research Letters
Volume13
Issue number4
Publication statusPublished - Jul 2006

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