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 language | English |
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Pages (from-to) | 145-169 |
Number of pages | 25 |
Journal | Discrete & Computational Geometry |
Volume | 60 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jul 2018 |
Externally published | Yes |
Keywords
- Lattice point
- Polytope
- Poisson summation
- Diophantine approximation