Normal Properties of Numbers in Terms of their Representation by the Perron Series

Abstract

We study the representation of real numbers by Perron series (P-representation) given by

$$\left(\left.0;1\right]\ni x=\sum_{n=0}^{\infty }\frac{{r}_{0}{r}_{1}\dots {r}_{n}}{\left({p}_{1}-1\right){p}_{1}\dots \left({p}_{n}-1\right){p}_{n}{p}_{n+1}}={\Delta }_{{p}_{1}{p}_{2}\dots }^{P}\right.,$$

where r_n, p_n ∈ ℕ, p_n+1 ≥ r_n + 1, and its transcoding (\(\overline{P }\)-representation)

$${x=\Delta }_{{g}_{1}{g}_{2}\dots }^{\overline{P} },$$

where 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 \({x=\Delta }_{{g}_{1}{g}_{2}\dots {g}_{2}\dots }^{\overline{P} }\) defined by the equality

$${\nu }_{i}^{\overline{P} }\left(x\right)=\underset{k\to \infty }{\mathrm{lim}}\frac{{N}_{i}^{\overline{P} }\left(x,k\right)}{k},$$

where \({N}_{i}^{\overline{P} }\left(x,k\right)\) denotes the amount of numbers n such that g_n = i and n ≤ k. In particular, we establish conditions under which the frequency \({\nu }_{i}^{\overline{P} }\left(x\right)\) exists and is constant for almost all x ∈ (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].