Inversion formulae for Siegel transforms

Abstract

Let $n\in {\mathbb{Z}}_{\ge 3}$ 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 $L^1({\mathbb{R}}^n)\cap L^2({\mathbb{R}}^n).$

The results in this note were already proved earlier by Ghosh--Kelmer--Yu. See Proposition 2.6 in the following paper: Anish Ghosh, Dubi Kelmer, Shucheng Yu, Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts, International Mathematics Research Notices, Volume 2022, Issue 6, March 2022, Pages 4682--4719, https://doi.org/10.1093/imrn/rnaa206