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.
Originalsprache | englisch |
---|---|
Seiten (von - bis) | 145-169 |
Seitenumfang | 25 |
Fachzeitschrift | Discrete & Computational Geometry |
Jahrgang | 60 |
Ausgabenummer | 1 |
DOIs | |
Publikationsstatus | Veröffentlicht - Juli 2018 |
Extern publiziert | Ja |