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 $$\nu =\delta{A_1}\delta_{A_2} * \cdots *\delta_{A_n}\cdots, $$ 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.