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Exact dimension functions of the prime continued fraction Cantor set
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 45 / Issue 6 / June 2025
- Published online by Cambridge University Press:
- 14 November 2024, pp. 1757-1776
- Print publication:
- June 2025
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- Article
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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 and Urbański (1999) and 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.
