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

TL;DR

This paper investigates the Hausdorff dimensions of specific sets of continued fractions with bounded odd and even-order partial quotients, revealing their size and structure through detailed mathematical analysis.

Contribution

It provides new results on the Hausdorff dimensions of sets of continued fractions with bounded and growing partial quotients, extending previous work on even-order cases.

Findings

Sum and product of certain continued fraction sets contain non-empty intervals

Hausdorff dimensions are computed for sets with exponential and super-exponential growth of partial quotients

Sets with bounded odd/even partial quotients have positive Hausdorff dimension

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.