On the number of lattice points in thin sectors

Ezra Waxman, Nadav Yesha

Research output: Contribution to journalArticlepeer-review

Abstract

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line y=αx with edges given by the lines y=(α±ϵ)x, where ϵ=ϵR→0 as R→∞. We establish an asymptotic count for Sα(ϵ,R), the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of ϵ and on the rationality/irrationality type of α. In particular, we demonstrate that if α is Diophantine, then Sα(ϵ,R) is asymptotic to the area of the sector, so long as ϵRt→∞ for some t<2.

Original languageEnglish
Pages (from-to)641-658
Number of pages18
JournalMonatshefte fur Mathematik
Volume204
Issue number3
DOIs
StatePublished - Jul 2024

Keywords

  • 11H06
  • 11P21
  • Diophantine
  • Lattice points
  • Sectors

All Science Journal Classification (ASJC) codes

  • General Mathematics

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