Lattice points in algebraic crosspolytopes 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 crosspolytopes 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 eprints
 Pub Date:
 August 2016
 DOI:
 10.48550/arXiv.1608.02417
 arXiv:
 arXiv:1608.02417
 Bibcode:
 2016arXiv160802417B
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 11K38;
 11P21 (Primary) 11J87 (Secondary)
 EPrint:
 27 pages