Macdonald's solid-angle sum for real dilations of rational polygons

TL;DR

This paper provides an explicit Fourier-analytic formula for the solid-angle sum of rational polygons under any real dilation, extending previous lattice-point enumeration results beyond integer dilations.

Contribution

The authors derive a new explicit formula for the solid-angle sum of rational polygons for real dilations using Fourier analysis, offering a novel perspective on lattice-point enumeration.

Findings

Explicit formula for $A_{ ext{P}}(t)$ for any real $t$ and rational polygons.

Enhanced understanding of lattice-point counting in real dilations.

Alternative approach to classical lattice-point enumeration methods.

Abstract

The solid-angle sum $A_{\mathcal{P}} (t)$ of a rational polytope ${\mathcal{P}} \subset \mathbb{R}^d$, with $t \in \mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\mathcal{P}} \subset \mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1992, but offers an alternate point of view for enumerating weighted lattice points in real dilations of real triangles.