Lattice points in algebraic cross-polytopes and simplices
Abstract
The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2016
- DOI:
- 10.48550/arXiv.1608.02417
- arXiv:
- arXiv:1608.02417
- Bibcode:
- 2016arXiv160802417B
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 11K38;
- 11P21 (Primary) 11J87 (Secondary)
- E-Print:
- 27 pages