Generalized Polya's theorem on connected locally compact Abelian groups of dimension 1

TL;DR

This paper extends Polya's theorem to a-adic solenoids, characterizing when the Gaussian distribution's properties hold in these more complex topological groups, using functional equations on their character groups.

Contribution

It provides a complete description of a-adic solenoids where the generalized Polya theorem applies, expanding the theorem's scope beyond the real line.

Findings

Characterization of a-adic solenoids satisfying the generalized Polya theorem

Reduction of the proof to solving functional equations in positive definite functions

Identification of conditions for the theorem's validity on these groups

Abstract

According to the generalized Polya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of a-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional equation in the class of continuous positive definite functions on the character group of an a-adic solenoid