Hausdorff dimension of sets with restricted, slowly growing partial quotients

TL;DR

This paper generalizes Hausdorff dimension results for sets of irrationals with partial quotients constrained by arbitrary subsets and slowly growing functions, establishing a precise dimension formula involving the exponent of convergence.

Contribution

It extends previous results by providing a dimension formula for sets with arbitrary restrictions on partial quotients and slow growth conditions.

Findings

Hausdorff dimension equals half the exponent of convergence of B

Generalizes Good's 1941 result to broader restrictions

Provides explicit dimension formula for complex partial quotient conditions

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 $a_n\geq f(n)$, 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 $B$.