Joint normality of representations of numbers: an ergodic approach

TL;DR

This paper introduces an ergodic approach to prove joint normality of number representations, demonstrating that almost all numbers are jointly normal with respect to different expansions like base-b and continued fractions.

Contribution

It develops a new ergodic framework for establishing joint normality of number representations and proves pointwise joint ergodicity for various number-theoretical maps.

Findings

Almost every number is jointly normal in base-b and continued fraction expansions.

Pointwise joint ergodicity holds for a wide class of number-theoretical maps.

Various forms of normality and joint normality are shown to be equivalent.

Abstract

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion and continued fraction expansion. This fact is a corollary of the following result which deals with {\em pointwise joint ergodicity}: Let $T_b:[0,1] \rightarrow [0,1]$ be the times $b$ map defined by $T_b x = bx \, \bmod \, 1 $ and let $T_G:[0,1] \rightarrow [0,1]$ be the Gauss map defined by $T_G(x) = \{\frac{1}{x}\}$ for $x \ne 0$ and $T_G (0) =0.$ (Here $\{ \cdot \}$ denotes the fractional part.) For any $f, g \in L^{\infty} (\lambda)$, \[ \lim_{N \rightarrow \infty} \frac{1}{N } \sum_{n=0}^{N-1} f(T_b^{n}x) \, g(T_G^n x) = \int f \, d \lambda \cdot \int g \, d \mu_G \quad \text{for almost every } x \in [0,1], \] where $\lambda$ is the Lebesgue measure on $[0,1]$ and $\mu_G$ is the Gauss measure on $[0,1]$ given by $\mu_G (A) = \frac{1}{ \log 2} \int_A \frac{1}{1+x} dx$ for any measurable set $A \subset [0,1]$. We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of number-theoretical maps of the interval and derive the corresponding corollaries pertaining to joint normality. We also establish the equivalence of various forms of normality and joint normality for representations of numbers, hereby providing a general framework for classical normality results.