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    Shparlinski, Igor 2008. Bilinear character sums over elliptic curves. Finite Fields and Their Applications, Vol. 14, Issue. 1, p. 132.

    Banks, William D. and Shparlinski, Igor E. 2007. Prime divisors in Beatty sequences. Journal of Number Theory, Vol. 123, Issue. 2, p. 413.

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Non-residues and primitive roots in Beatty sequences

  • William D. Banks (a1) and Igor E. Shparlinski (a2)
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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.

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COPYRIGHT: © Australian Mathematical Society 2006
References
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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