Ergodic Abelian actions with homogeneous spectrum

TL;DR

This paper constructs weakly mixing actions of certain Abelian groups that have a homogeneous spectrum of any prescribed finite multiplicity, expanding the understanding of spectral properties in ergodic theory.

Contribution

It demonstrates the existence of weakly mixing probability-preserving actions with homogeneous spectrum of arbitrary finite multiplicity for a broad class of Abelian groups.

Findings

Existence of actions with homogeneous spectrum of any finite multiplicity

Applicable to all infinite countable discrete Abelian groups

Extends spectral theory in ergodic group actions

Abstract

It is shown that for each $N>0$ and for a wide class of Abelian non-compact locally compact second countable groups $G$ including all infinite countable discrete ones and $\Bbb R^{d_1}\times\Bbb Z^{d_2}$ with $d_1,d_2\ge 0$, there exists a weakly mixing probability preserving $G$-action with a homogeneous spectrum of multiplicity $N$.