Transcendence and measures via the refined Diophantine exponent

Abstract

In 2007, Adamczewski and Bugeaud introduced the notion of the Diophantine exponent of an infinite word as a quantitative measure of repetition, leading to new transcendence results for real numbers whose expansions in an integer base are sufficiently simple. In the present article, we introduce the refined Diophantine exponent, which detects weaker forms of repetition while preserving the full strength of the classical approach. This new exponent applies in situations where repetition is partially obscured by some form of noise. Related ideas already appear in the work of Corvaja and Zannier in 2002 and, more recently, in the works of Kebis, Luca, Ouaknine, Scoones, and Worrell. Our approach provides a unified framework that recovers and extends these results, as well as those of Adamczewski and Bugeaud. We also develop quantitative refinements of this method, leading to results about transcendence measures. The recent breakthrough of Bell, Diller, and Jonsson in the context of algebraic dynamics is partly based on a similar idea, which also served as a motivation for the present work.