Weak Convergence and Spectrality of Infinite Convolutions

TL;DR

This paper establishes conditions for the convergence of infinite convolutions of finite sets, explores their spectral properties, constructs singular spectral measures, and analyzes the dimensional properties of their supports.

Contribution

It provides a necessary and sufficient condition for the existence of infinite convolutions and investigates spectrality and dimensional properties of associated measures.

Findings

Established a criterion for infinite convolution convergence.

Constructed singular spectral measures without compact support.

Showed the abundance and intermediate-value property of support dimensions.

Abstract

Let $\{ A_k\}_{k=1}^\infty$ be a sequence of finite subsets of $\mathbb{R}^d$ satisfying that $\# A_k \ge 2$ for all integers $k \ge 1$. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution $$\nu =\delta_{A_1}*\delta_{A_2} * \cdots *\delta_{A_n}*\cdots, $$ where all sets $A_k \subseteq \mathbb{R}_+^d$ and $\delta_A = \frac{1}{\# A} \sum_{a \in A} \delta_a$. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in $\mathbb{R}$ and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.