2018/04/09 by Christoph Aistleitner, Aistleitner, Christoph, Veronica Becher +3
Computer Science · Mathematics · #11K16 #68R15 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Number Theory (math.NT) #Probability (math.PR) #cs.FL #math.CO #math.NT #math.PR #msc:11K16 #msc:68R15
paper · pdf · doi:10.48550/arxiv.1804.02844
21 pages. This paper will appear in Trans. AMS
arxiv created 2018/09/17 · arxiv updated 2018/09/18
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, still, 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 TP, which is the set of all sequences a1a2 … of symbols from \0, …, b-1\ such that an is equal to apn, for every p in P and n=1,2,…. 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 TP 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 TP is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set TP and we characterize the normal sequences whose Toeplitz transform is normal as well.