Bulletin of the Australian Mathematical Society

Research Article

Non-residues and primitive roots in Beatty sequences

William D. Banksa1 and Igor E. Shparlinskia2

a1 Department of Mathematics, University of Missouri, Columbia, MO 65211, United States of America, e-mail: bbanks@math.missouri.edu

a2 Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequenceS0004972700035449_inline1 where α,β xs2208 xs211D, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p½+εNp, then among the first N elements of xs212Cα,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) admits a sharper estimate.

(Received January 31 2006)