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arXiv · 2608.20191

Spectrum of the refined Diophantine exponent

Quang-Khai Nguyen
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Abstract

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin--Shapiro, and Thue--Morse words, words coming from coding a rotation by intervals, and bracket words.

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Quang-Khai Nguyen. 2026-08-20. Spectrum of the refined Diophantine exponent. https://arxiv.org/abs/2608.20191

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