Tushar Das, Mathematics & Statistics, authored the article "Exact dimension functions of the prime continued fraction Cantor set" in Ergodic Theory and Dynamical Systems published on Monday, May 5 by Cambridge University Press. In joint work with David Simmons (York), we prove that the Hausdorff measure of the prime Cantor set, which comprises the irrationals whose continued fraction entries are prime numbers, cannot be finite and positive with respect to any sufficiently regular gauges, thus negatively answering questions of Mauldin and Urbański (1999) and Mauldin (2013). By contrast, assuming a conjecture on prime gaps that extends the Cramér–Granville heuristics, we prove that the packing measure of the conformal measure on the prime Cantor set is positive and finite with respect to the gauge 𝜓(𝑟)=𝑟^𝛿 log^(−2𝛿)log(1/𝑟), where 𝛿 is the fractal dimension of the prime Cantor set.
Submitted on: May 5