Discrepancy of rational points in simple algebraic groups

TL;DR

This paper investigates the distribution irregularities of rational points on semisimple algebraic groups, providing explicit discrepancy estimates and an analogue of Schmidt's theorem for counting solutions to Diophantine inequalities.

Contribution

It introduces new discrepancy bounds for rational points on algebraic groups and extends Schmidt's theorem to this setting with effective asymptotic counting.

Findings

Derived mean-square, almost sure, and uniform discrepancy estimates with explicit error bounds.

Established an analogue of Schmidt's theorem for rational points on group varieties.

Provided effective bounds for counting rational solutions to Diophantine inequalities.

Abstract

The present paper analyzes the discrepancy of distribution of rational points on general semisimple algebraic group varieties. The results include mean-square, almost sure, and uniform discrepancy estimates with explicit error bounds, which apply to general families of subsets, and are valid at arbitrarily small scales. We also consider an analogue of W. Schmidt's classical theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in Euclidean spaces. We formulate and prove a version of it for rational points on the group variety, together with an effective bound which in some instances can be expected to be best possible.