Normal numbers with digit dependencies

TL;DR

This paper establishes that most real numbers are normal even with certain digit dependencies, especially in Toeplitz sets, and characterizes when normality is preserved under Toeplitz transformations.

Contribution

It provides metric theorems quantifying digit dependence limits for normality and analyzes normality in Toeplitz sets, including cases with multiple prime dependencies.

Findings

Almost all real numbers are normal with slight digit dependence.

In Toeplitz sets with prime dependencies, most numbers are normal to base b.

For singleton prime sets, numbers are normal in all bases.

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_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.