Lattice Points in Algebraic Cross-polytopes and Simplices

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Abstract

The number of lattice points |tP∩Zd|, as a function of the real variable t>1 is studied, where P⊂Rd belongs to a special class of algebraic cross-polytopes and simplices. It is shown that the number of lattice points can be approximated by an explicitly given polynomial of t depending only on P. The error term is related to a simultaneous Diophantine approximation problem for algebraic numbers, as in Schmidt’s theorem. The main ingredients of the proof are a Poisson summation formula for general algebraic polytopes, and a representation of the Fourier transform of the characteristic function of an arbitrary simplex in the form of a complex line integral.
Original languageEnglish
Pages (from-to)145-169
Number of pages25
JournalDiscrete & Computational Geometry
Volume60
Issue number1
DOIs
Publication statusPublished - Jul 2018
Externally publishedYes

Keywords

  • Lattice point
  • Polytope
  • Poisson summation
  • Diophantine approximation

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