Weak Convergence and Spectrality of Infinite Convolutions

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 $ν=δ_{A_1}*δ_{A_2} * \cdots *δ_{A_n}*\cdots,$ where all sets $A_k \subseteq \mathbb{R}_+^d$ and $δ_A = \frac{1}{\# A} \sum_{a \in A} δ_a$. 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.

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