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

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 Hančl 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.