On the $b$-ary expansions of $\log (1 + \frac{1}{a})$ and ${\mathrm e}$

TL;DR

This paper explores the relationship between the irrationality exponent of certain real numbers and the complexity of their base-$b$ expansions, revealing that numbers with irrationality exponent close to 2 have non-trivial digit structures.

Contribution

It establishes a novel link between the irrationality exponent and the complexity of $b$-ary expansions for classical numbers like logarithms and powers of $e$, under specific conditions.

Findings

Numbers with irrationality exponent 2 or slightly greater have complex $b$-ary expansions.

The result applies to badly approximable numbers, powers of $e$, and certain logarithms.

It uncovers an unexpected connection between irrationality measures and digit expansion complexity.

Abstract

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We prove that, if the irrationality exponent of $\xi$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of ${\mathrm e}$, and $\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its $b$-ary expansion.