New families of normal numbers constructed from existing ones

Abstract. Given an integer \(q\ge 2\), let \({\cal A}_q:=\{0,1,\ldots,q-1\}\) be the set of base \(q\) digits. We say that \(\alpha=a_1\ldots a_k\), where each \(a_i \in {\cal A}_q\), is a \({\it word}\) of length \(\lambda(\alpha)=k\). Let \(H(1),H(2),\ldots\) be a sequence of nonnegative integers and given a sequence of words \(\alpha_1, \alpha_2,\ldots\), we examine under which conditions the number \(0.\alpha_1^{H(1)} \alpha_2^{H(2)} \ldots\) is a normal number in base \(q\) (where \(\alpha_j^r=\underbrace{\alpha_j \ldots \alpha_j}_{r\ \rm{times}}\)).