Joint normality of representations of numbers: an ergodic approach

Abstract

We introduce an ergodic approach to the study of joint normality of representations of numbers.For example, we show that for any integer b ≥ 2 almost every number x ∈ [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 pointwise joint ergodicity:where λ is the Lebesgue measure on [0, 1] and µ G is the Gauss measure on [0, 1] given by µ G (A) =We show that the phenomenon of the pointwise joint ergodicity takes place for a wide variety of numbertheoretical 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. VITALY BERGELSON AND YOUNGHWAN SONNowadays this result can be easily obtained by means of the ergodic theory. 2 Indeed, consider the map