Hausdorff dimension of sets with restricted, slowly growing partial quotients

Abstract

I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\ge f(n)$, where $B$ is an infinite subset of $\mathbb{N}$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty )$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that \[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in \mathbb{N}$, and }a_n\to\infty\text{ as }n\to\infty\] is of Hausdorff dimension $\tau (B)/2,$ where $\tau (B)$ is the exponent of convergence of $B$.

arXiv commentto appear in Proc. Amer. Math. Soc