On the expansion of some exponential periods in an integer base

TL;DR

This paper establishes a lower bound on the subword complexity of base-$b$ expansions for certain real numbers, including $e$, generalizing previous results and advancing understanding of their combinatorial properties.

Contribution

It provides the first lower bound for the subword complexity of $e$ and similar transcendental exponential periods, extending prior theoretical frameworks.

Findings

Lower bound for subword complexity of base-$b$ expansions derived

Generalization of Ferenczi and Mauduit's theorem achieved

First bounds established for $e$ and related transcendental numbers

Abstract

We derive a lower bound for the subword complexity of the base-$b$ expansion ($b\geq 2$) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to Ferenczi and Mauduit. As a consequence, we obtain the first lower bound for the subword complexity of the number $e$ and of some other transcendental exponential periods.