Normal numbers with digit dependencies

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, 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 T P , which is the set of all sequences a 1 a 2 . . . 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 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.

Normal sequences with given limits of multiple ergodic averages

“…The last statement of Theorem 1.3 was recently proved, in higher generality, by Aistleitner, Becher, and Carton [1].…”

On simply normal numbers with digit dependencies

“…is an element of T P . Motivated by the question posed in [1] on how to exhibit a normal number in T P for any set P of primes, we construct in this note simply normal numbers for arbitrary bases and a large family of sets P.…”

“…In [1], Aistleitner, Becher, and Carton considered the notion of Borel normality under the assumption of dependencies between the digits of the expansion. Thus, [1, theorem 1] states that, given any integer base $$b \geqslant 2$$ and any finite subset $${\EuScript P}$$ of the primes, almost all numbers, with respect to the uniform probability measure on $${\EuScript T}_{\EuScript P}$$, are normal to the base b .…”

“…In [1], Aistleitner, Becher, and Carton considered the notion of Borel normality under the assumption of dependencies between the digits of the expansion. Thus, [1, theorem 1] states that, given any integer base $$b \geqslant 2$$ and any finite subset $${\EuScript P}$$ of the primes, almost all numbers, with respect to the uniform probability measure on $${\EuScript T}_{\EuScript P}$$, are normal to the base b . In the particular case $${\EuScript P}= \lbrace 2\rbrace$$, they show [1, theorem 2] that almost all elements in $${\EuScript T}_{\EuScript P}$$ (still with respect to the uniform measure on $${\EuScript T}_{\EuScript P}$$) are normal to all integer bases greater than or equal to 2.…”

“…Thus, [1, theorem 1] states that, given any integer base $$b \geqslant 2$$ and any finite subset $${\EuScript P}$$ of the primes, almost all numbers, with respect to the uniform probability measure on $${\EuScript T}_{\EuScript P}$$, are normal to the base b . In the particular case $${\EuScript P}= \lbrace 2\rbrace$$, they show [1, theorem 2] that almost all elements in $${\EuScript T}_{\EuScript P}$$ (still with respect to the uniform measure on $${\EuScript T}_{\EuScript P}$$) are normal to all integer bases greater than or equal to 2. For $${\EuScript P}=\lbrace 2\rbrace$$, a construction of an explicit number in $${\EuScript T}_{\EuScript P}$$ that is normal to the base 2 appears in [2].…”