Integrability of Siegel transforms and an application

TL;DR

This paper establishes algebraic criteria for the integrability of a generalized Siegel transform in the context of rational representations of semisimple algebraic groups, with applications to counting rational approximations.

Contribution

It provides new algebraic criteria for $L^p$-integrability of generalized Siegel transforms and applies these results to effective counting of rational points on flag varieties.

Findings

Established sharp algebraic criteria for $L^p$-integrability of generalized Siegel transforms.

Derived an effective asymptotic formula for counting rational approximations.

Connected integrability conditions with equidistribution estimates for group orbits.

Abstract

We establish sharp algebraic criteria for the $L^{p}$-integrability, for $p = 1, 2, \infty$, of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic $\mathbb{Q}$-groups, extending Siegel's analytic work in the geometry of numbers. As an application, we derive an effective asymptotic formula for the number of rational approximations of bounded height to almost every real point on a rank-one flag variety at the Diophantine exponent. The argument combines the integrability criterion with effective equidistribution estimates for translated orbits of maximal compact subgroups, a result of independent interest.