Strong wavefront lemma and counting lattice points in sectors

TL;DR

This paper introduces the strong wavefront lemma to analyze the asymptotic count of lattice points in sectors of affine symmetric spaces, advancing understanding of quadratic forms and lattice distributions.

Contribution

It presents the strong wavefront lemma, a novel tool proving uniform Lipschitz properties of the generalized Cartan decomposition in symmetric spaces.

Findings

Asymptotic formulas for counting integral quadratic forms with specific decompositions

Asymptotic estimates for lattice points in sectors of affine symmetric spaces

Establishment of the strong wavefront lemma as a key analytical tool

Abstract

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.