On the Longest Block Function in Continued Fractions

Abstract

For an irrational number $x\in [0,1)$ , let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$ . Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$ , for $n\geq 1$ , the $n$ th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$ , which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$ . We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

On the Shortest Distance Function in Continued Fractions

“…Wang and Wu [7] considered the metrical properties of the maximal run-length function which counts the longest run of the same symbol among the first n partial quotients of x . They proved that, for almost all Song and Zhou [6] gave a more subtle characterisation of the function In this paper, we continue the study by considering the shortest distance function This is motivated by the behaviour of the shortest distance between two orbits, in the continued fraction system. Shi et al [5] proved that, for almost all where is the Rényi entropy defined by (1.2).…”

“…Song and Zhou [6] gave a more subtle characterisation of the function R n (x). In this paper, we continue the study by considering the shortest distance function…”