Normal numbers with digit dependencies

• Published in: Transactions of the American Mathematical Society 2019-09, Vol.372 (6), p.4425-4446
• Main Authors: Aistleitner, Christoph, Becher, Verónica, Carton, Olivier
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7706

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 \log \log n consecutive digits with indices starting at position n are independent. As the main application, we consider the Toeplitz set T_P, which is the set of all sequences a_1a_2 \ldots of symbols from \{0, \ldots , b-1\} such that a_n is equal to a_{pn} for every p in P and n=1,2,\ldots . Here b is an integer base and P is a finite set of prime numbers. We show that almost every real number whose base b expansion is in T_P is normal to base b. In the case when P is the singleton set \{2\} we prove that more is true: almost every real number whose base b expansion is in T_P is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set T_P, and we characterize the normal sequences whose Toeplitz transform is normal as well.