Weak Convergence and Spectrality of Infinite Convolutions

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 $$\nu ={\delta }_{A_1}\ast {\delta }_{A_2}\ast \cdots \ast {\delta }_{A_n}\ast \cdots ,$$ 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.

23 pages