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arXiv · 2507.15552

Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients

Yuefeng Tang
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Abstract

We study the continued fractions with bounded odd/even-order partial quotients. In particular, we investigate the sizes of the sets of continued fractions whose odd-order partial quotients are equal to 1. We demonstrate that the sum and the product of two sets of continued fractions whose odd-order partial quotients are equal to 1 both contain non-empty intervals. Our work compliments the results of Han\v{c}l and Turek on the set of continued fractions whose even-order partial quotients are equal to 1. Furthermore, we determine the Hausdorff dimensions of the sets of continued fractions whose odd-order partial quotients are equal to 1 and even-order partial quotients are growing at an exponential rate, a super-exponential rate, and in general a positive function rate.

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Yuefeng Tang. 2025-07-21. Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients. https://doi.org/10.1007/s11139-025-01288-z

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