Normal properties of numbers in the terms of their representation by the Perron series

Abstract

UDC 511.7

We examine the representation of real numbers by the Perron series ($P$-representation) given by $$(0;1]\ni x=\sum\limits_{n=0}^{\infty}\frac{r_0 r_1\ldots r_n}{(p_1-1)p_1\ldots(p_n-1)p_n p_{n+1}}=\Delta^P_{p_1 p_2\ldots},\quad\mbox{where}\quad r_n,p_n\in\mathbb{N},\quad p_{n+1}\geq r_n+1,$$ and its transcoding ($\overline{P}$-representation) $$x=\Delta^{\overline{P}}_{g_1 g_2\ldots},\quad\mbox{where}\quad g_n=p_n-r_{n-1}.$$ We establish the properties of $\overline{P}$-representations typical of almost all numbers with respect to the Lebesgue measure (normal properties of the representations of numbers). We also examine the conditions of existence of the frequency of a digit $i$ in the $\overline{P}$-representation of a number $x=\Delta^{\overline{P}}_{g_1 g_2\ldots g_n\ldots}$ defined by the equality $$\nu_i^{\overline{P}}(x)=\lim\limits_{k\to\infty} \frac{N^{\overline{P}}_i(x,k)}{k},$$ where $N^{\overline{P}}_i(x,k)$ denotes the amount of numbers $n$ such that $g_n=i$ and $n\leq k.$ In particular, we establish conditions under which the frequency $\nu_i^{\overline{P}}(x)$ exists and is constant for almost all $x\in(0;1].$ In addition, we also determine the conditions under which the digits in $\overline{P}$-representations are encountered finitely or infinitely many times for almost all numbers from $(0;1].$