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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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  • Bulletin of the Australian Mathematical Society, Volume 73, Issue 3
  • June 2006, pp. 433-443

Non-residues and primitive roots in Beatty sequences

  • William D. Banks (a1) and Igor E. Shparlinski (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700035449
  • Published online: 01 April 2009
Abstract

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+εNp, 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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[5]J. Bourgain and S.V. Konyagin , ‘Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order’, C. R. Math. Acad. Sci. Paris 337 (2003), 7580.

[7]A.S. Fraenkel and R. Holzman , ‘Gap problems for integer part and fractional part sequences’, J. Number Theory 50 (1995), 6686.

[9]D.R. Heath-Brown and S.V. Konyagin , ‘New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sum’, Quart. J. Math. 51 (2000), 221235.

[11]A.Y. Khinchin , ‘Zur metrischen Theorie der diophantischen Approximationen’, Math. Z. 24 (1926), 706714.

[13]T. Komatsu , ‘The fractional part of nϑ + φ and Beatty sequences’, J. Théor. Nombres Bordeaux 7 (1995), 387406.

[18]K. O'Bryant , ‘A generating function technique for Beatty sequences and other step sequences’, J. Number Theory 94 (2002), 299319.

[24]W.M. Schmidt , Diophantine approximation (Springer-Verlag, Berlin, 1980).

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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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