Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
Number Theory
2026-01-21 v1 Dynamical Systems
Abstract
Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions.
Keywords
Cite
@article{arxiv.2601.12737,
title = {Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions},
author = {Yuto Nakajima and Hiroki Takahasi and Baowei Wang},
journal= {arXiv preprint arXiv:2601.12737},
year = {2026}
}
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20 pages