Typicality of normal numbers with respect to the Cantor series expansion
Let \(Q=\{q_n\}_{n=1}^\infty\) be a sequence of integers. If each \(q_n\geq2\) then we call \(Q\) a basic sequence. Then the \(Q\)-Cantor series expansion of a real \(x\in[0,1)\) is the expansion of the form NEWLINE\[NEWLINE x=\sum_{n=1}^\infty\frac{E_n}{q_1q_2\cdots q_n} NEWLINE\]NEWLINE such that each \(E_n\) is taken from the set \(\{0,1,\dots,q_n-1\}\) with \(E_n\not=q_n-1\) infinitely often. By setting \(q_n=q\) for \(n\geq1\) one gets the \(q\)-adic expansion.NEWLINENEWLINEIn the present paper the author investigates three different notions of normality. Let \(N_n^Q(B,x)\) denote the number of occurrences of a block \(B\) starting at a position no greater than \(n\) in the \(Q\)-Cantor series expansion of \(x\). Additionally, we define NEWLINE\[NEWLINE Q_n^{(k)}=\sum_{j=1}^n\frac{1}{q_jq_{j+1}\cdots q_{j+k-1}}. NEWLINE\]NEWLINE Then we call a real number \(x\in[0,1)\) \(Q\)-normal if for \(k\geq1\) and all blocks \(B\) of length \(k\) we have NEWLINE\[NEWLINE \lim_{n\to\infty}\frac{N_n^Q(B,x)}{Q_n^{(k)}}=1. NEWLINE\]NEWLINE The second notion of strongly normal numbers does not count the occurrences at all positions but only at those in an arithmetic progression. Finally the third notion of ratio normal numbers investigates the ration of the occurrences of different blocks.NEWLINENEWLINEThe author investigates the typicality of numbers with respect to these three notions. In particular, among other things he shows the following. If \(Q\) is a basic sequence such that \(\lim_{n\to\infty}q_n=\infty\) and \(B\) is a block of length \(k\), then for almost all real numbers \(x\in[0,1)\), one has NEWLINE\[NEWLINEN_n^Q(B,x)=Q_n^{(k)}+\mathcal{O}\left(\sqrt{Q_n^{(k)}}\left(\log\log Q_n^{(k)}\right)^{\frac12}\right).NEWLINE\]NEWLINE This extends former results of \textit{F. Schweiger} [Monatsh. Math. 74, 150--153 (1970; Zbl 0188.35103)] and \textit{J. Galambos} [Representations of real numbers by infinite series. Berlin etc.: Springer (1976; Zbl 0322.10002)].
scientific article; zbMATH DE number 6032688
scientific article