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arXiv · 2601.12737

Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions

Yuto Nakajima·Hiroki Takahasi·Baowei Wang
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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.

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Yuto Nakajima, Hiroki Takahasi, Baowei Wang. 2026-01-19. Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions. https://arxiv.org/abs/2601.12737

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