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.