New spectral multiplicities for ergodic actions

TL;DR

This paper demonstrates that for certain Abelian group actions, any specified multiplicative or additive subsemigroup of natural numbers can be realized as the spectral multiplicity set of a weakly mixing ergodic action.

Contribution

It constructs explicit examples of ergodic actions with prescribed spectral multiplicity sets for Abelian groups, extending previous understanding of spectral multiplicities.

Findings

Realization of specific multiplicity sets for G-actions

Construction of weakly mixing ergodic actions with desired spectral properties

Extension of spectral multiplicity realization to broader classes of groups

Abstract

Let G be a locally compact second countable Abelian group. Given a measure preserving action T of G on a standard probability space, let M(T) denote the set of essential values of the spectral multiplicity function of the Koopman unitary representation of G associated with T. In the case when G is either a discrete countable Abelian group or R^n, n>0, it is shown that the sets of the form {p,q,pq}, {p,q,r,pq,pr,qr,pqr} etc. or any multiplicative (and additive) subsemigroup of N are realizable as M(T) for a weakly mixing G-action T.