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Non-residues and primitive roots in Beatty sequences

Published online by Cambridge University Press:  17 April 2009

William D. Banks  and
Igor E. Shparlinski
William D. Banks
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, United States of America, e-mail: bbanks@math.missouri.edu
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
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We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence 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 p½+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) admits a sharper estimate.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 3 , June 2006 , pp. 433 - 443
Copyright
Copyright © Australian Mathematical Society 2006

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