PreprintarXiv licencearXiv
Spectrum of the refined Diophantine exponent
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,∞]. 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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BibTeX
@misc{nguyen2026spectrum,
title = {{Spectrum of the refined Diophantine exponent}},
author = {Quang-Khai Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.20191},
eprint = {2608.20191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20191},
}
RIS
TY - UNPB
TI - Spectrum of the refined Diophantine exponent
AU - Quang-Khai Nguyen
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.20191
UR - https://arxiv.org/abs/2608.20191
AB - 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,∞]. 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.
ER -