Analysis reveals new insights into Lebesgue measure and dimensions in real sets with large prime quotients, indicating significant extensions of prior theorems.
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 Łuczak (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{ü}ller (2023).
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Robert et al. (2025) studied this question.
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