Autor(es)

Christoph Aistleitner and Verónica Becher and Olivier Carton

URL Pública

Abstract

We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than consecutive digits with indices starting at position are independent. As the main application, we consider the Toeplitz set , which is the set of all sequences of symbols from such that is equal to for every in and . Here is an integer base and is a finite set of prime numbers. We show that almost every real number whose base expansion is in is normal to base . In the case when is the singleton set we prove that more is true: almost every real number whose base expansion is in is normal to all …