On the maximal run-length function in continued fractions

Abstract. This paper is concerned with the metrical property and fractal structure of maximal run-length function in an infinite symbolic system: continued fraction dynamical system. More precisely, let \([a_1(x),a_2(x),\ldots]\) be the continued fraction expansion of \(x\in [0,1)\). Call $$ R_n(x):=\max_{i\geq 1}\big\{k: a_{j+1}(x)=\cdots=a_{j+k}(x)=i, \ {\rm{for \ some}}\ 0\leq j\leq n-k\big\} $$ the \(n\)-th maximal run-length function of \(x\), which represents the longest run of same symbol in the first \(n\) partial quotients of \(x\). We show that $$ \lim_{n\to \infty}\frac{R_n(x)}{\log_{\frac{\sqrt{5}+1}{2}}n}=\frac{1}{2}, \ \ \text{a.e.} \ x\in [0,1). $$ This extends a result of Erdős and Rényi in finite symbolic space. At the same time, fractal structure of exceptional sets with respect to above metrical result are also studied.

Key words and phrases. Maximal run-length function, continued fractions, metrical theory, Hausdorff dimension.