Lattice Points in Algebraic Cross-polytopes and Simplices

Abstract

The number of lattice points \(| tP \cap \mathbb {Z}^d |\), 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.