Inversion formulae for Siegel transforms

TL;DR

This paper establishes Lebesgue-almost everywhere pointwise inversion formulas for Siegel transforms in the geometry of numbers, applicable to a broad class of functions including even, compactly supported, and integrable functions.

Contribution

It provides general inversion formulas for Siegel transforms valid for a wide class of functions, extending previous results in the geometry of numbers.

Findings

Valid for any even, compactly supported, and integrable functions

Applicable to functions in both L^1 and L^2 spaces

Provides Lebesgue-almost everywhere pointwise inversion formulas

Abstract

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for the Siegel transforms of any even and compactly supported Borel measurable function $f : \mathbb{R}^n \to \mathbb{R}$ that belongs to $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n).$