Spectrality and supports of infinite convolutions in $\mathbb{R}^d$

TL;DR

This paper extends the understanding of spectral measures and their supports for infinite convolutions in higher-dimensional spaces, providing new conditions for compact support and exploring fractal dimensions of these measures.

Contribution

It generalizes previous results from one-dimensional to multi-dimensional spaces, offering new criteria for the existence of compact supports and analyzing the fractal properties of infinite convolutions.

Findings

Characterization of spectral measures with and without compact supports in $\\mathbb{R}^d$

Necessary and sufficient conditions for infinite convolutions to have compact supports

Existence of spectral measures with arbitrary Hausdorff and packing dimensions

Abstract

We study the spectrality of a class of infinite convolutions in $\mathbb{R}^d$, generalizing a result given by Li, Miao and Wang in 2022 from $\mathbb{R}$ to $\mathbb{R}^d$. This allows us to easily construct spectral measures with and without compact supports in $\mathbb{R}^d$, and motivates us to systematically study the supports of infinite convolutions. In particular, we give a sufficient and necessary condition for infinite convolutions to exist with compact supports, generalizing a related well-known result which is widely used. After giving strong relations between supports of infinite convolutions and sets of infinite sums, we study the closedness and fractal dimensions of infinite sums of union sets in order to deal with non-compact supports of infinite convolutions. As an application of these new tools, we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in $\mathbb{R}^d$, generalizing another result given by Li, Miao and Wang in 2022 from $\mathbb{R}$ to $\mathbb{R}^d$.