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<p>We consider the distribution of perfect squares in a Beatty sequence, which is of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_998_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lfloor \alpha n +\beta \rfloor \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mi>α</mi> <mi>n</mi> <mo>+</mo> <mi>β</mi> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation>. We also obtain a new bound for character sums along Beatty sequences with composite moduli, which answers an open question posted by Igor Shparlinski.</p>

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Squares in a Beatty sequence and an improvement on character sums

  • 期刊论文

  • 发表日期：

    2025年2月4日

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  • Victor Zhenyu Guo,
  • Jinyun Qi,
  • Mengyao Jing

摘要

We consider the distribution of perfect squares in a Beatty sequence, which is of the form \(\lfloor \alpha n +\beta \rfloor \) ⌊ α n + β ⌋ . We also obtain a new bound for character sums along Beatty sequences with composite moduli, which answers an open question posted by Igor Shparlinski.