Lattice points in algebraic cross-polytopes and simplices

TL;DR

This paper investigates the lattice point count in algebraic cross-polytopes and simplices, providing an explicit polynomial approximation and linking the error to Diophantine approximation, with key techniques involving Poisson summation and Fourier analysis.

Contribution

It introduces a polynomial approximation for lattice points in algebraic polytopes and relates the approximation error to algebraic Diophantine approximation, using novel Fourier analysis methods.

Findings

Lattice point count approximated by an explicit polynomial in t

Error term connected to algebraic Diophantine approximation

Poisson summation formula adapted for algebraic polytopes

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.