Well-distribution of Polynomial maps on locally compact groups

TL;DR

This paper extends classical equidistribution results to polynomial maps from locally compact groups into compact abelian groups, generalizing prior work on polynomial sequences in nilmanifolds.

Contribution

It formulates and proves new equidistribution theorems for polynomial maps between locally compact and compact abelian groups, broadening the scope of previous results.

Findings

Established general equidistribution theorems for polynomial maps

Extended classical results to broader group settings

Provided new insights into distribution properties of polynomial maps

Abstract

Weyl's classical equidistribution theorem states that real-valued polynomial sequences are uniformly distributed modulo 1, unless all non-constant coefficients are rational. A continuous function between two topological groups is called a \emph{polynomial map} of degree at most $d$ if it vanishes under any $d+1$ difference operators. Leibman, and subsequently Green and Tao, formulated and proved equidistribution theorems about polynomial sequences that take values in a nilmanifold. We formulate and prove some general equidistribution theorems regarding polynomial maps from a locally compact group into a compact abelian group.