Fractional dimension of some exceptional sets in continued fractions

Abstract

In this paper, we calculate the Hausdorff dimension of some exceptional sets that emerge from specific constraints imposed on the partial quotients of continued fractions. In particular, we calculate the Hausdorff dimension of the sets \[ \Lambda _1=\Biggl \lbrace x\in (0,1): a_{n+1}(x)\ge \sum _{i=1}^n a_i(x), \text{ for all } n\in \mathbb{N}\Biggr \rbrace , \] and \[ \Lambda _2=\Biggl \lbrace x\in (0,1): a_{n+1}(x)\ge \sum _{i=1}^n a_i(x), \text{ for infinitely many } n\in \mathbb{N}\Biggr \rbrace . \] We prove that the Hausdorff dimensions of $\Lambda _1$ and $\Lambda _2$ are $1/2$ and $1$ respectively. The Hausdorff dimension of some other related sets, obtained by considering different faster growth rates such as replacing the growth rate of sums of partial quotients with the product of partial quotients in the above sets, is also calculated with the dimension bounds $1/3$ and at least $2/3$.

Keywords

• Continued fractions
• growth rate
• Hausdorff dimension