Well-distribution of Polynomial maps on locally compact groups

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.