Equidistribution of Dense Subgroups on Nilpotent Lie Groups

TL;DR

This paper proves that dense subgroups of simply connected nilpotent Lie groups become uniformly distributed across the entire group as the word metric radius increases, with explicit rates and random walk analogues.

Contribution

It establishes equidistribution of dense subgroups in nilpotent Lie groups with quantitative rates and extends results to random walk averages.

Findings

$S_n$ becomes equidistributed with Haar measure as n increases.

Provides explicit convergence rates for equidistribution.

Proves a local limit theorem for random walk averages.

Abstract

Let $\Gamma$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $\Gamma$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem).