Counting intrinsic Diophantine approximations in simple algebraic groups

TL;DR

This paper derives an explicit asymptotic formula for counting rational solutions to intrinsic Diophantine inequalities within simple algebraic groups at very small scales, advancing understanding in Diophantine approximation.

Contribution

It provides the first explicit asymptotic count of rational solutions in simple algebraic groups for small-scale inequalities, a significant step in Diophantine approximation theory.

Findings

Explicit asymptotic formula for rational solutions

Counts solutions at arbitrarily small scales

Advances understanding of Diophantine inequalities in algebraic groups

Abstract

We establish an explicit asymptotic formula for the number of rational solutions of intrinsic Diophantine inequalities on simply-connected simple algebraic groups, at arbitrarily small scales.