Exact dimensions of the prime continued fraction Cantor set

Abstract

We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$ $set$, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi(r) = r^\delta \log^{-2\delta}\log(1/r)$, where $\delta$ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.