Continued fractions with large prime partial quotients

TL;DR

This paper investigates the measure and dimension of real numbers with infinitely many large prime partial quotients in their continued fraction expansions, extending previous theorems and deriving new asymptotics for related zeta functions.

Contribution

It extends classical results by analyzing sets with large prime partial quotients and introduces new asymptotics for the almost prime zeta function.

Findings

Determines Lebesgue measure and Hausdorff dimension of specific sets of real numbers.

Provides new asymptotic formulas for the tail of the almost prime zeta function.

Includes recent results by Schindler-Zweimüller (2023).

Abstract

We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by {\L}uczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweim{\"u}ller (2023).