Maximal run-length function with constraints: a generalization of the Erd\H{o}s-R\'enyi limit theorem and the exceptional sets

Abstract

Let $\mathbf{A}=\{A_i\}_{i=1}^{\infty}$ be a sequence of sets with each $A_i$ being a non-empty collection of $0$-$1$ sequences of length $i$. For $x\in [0,1)$, the maximal run-length function $\ell_n(x,\mathbf{A})$ (with respect to $\mathbf{A}$) is defined to the largest $k$ such that in the first $n$ digits of the dyadic expansion of $x$ there is a consecutive subsequence contained in $A_k$. Suppose that $\lim_{n\to\infty}(\log_2|A_n|)/n=\tau$ for some $\tau\in [0,1]$ and one additional assumption holds, we prove a generalization of the Erd\H{o}s-R\'enyi limit theorem which states that \[\lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=\frac{1}{1-\tau}\] for Lebesgue almost all $x\in [0,1)$. For the exceptional sets, we prove under a certain stronger assumption on $\mathbf{A}$ that the set \[\left\{x\in [0,1): \lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=0\text{ and } \lim_{n\to\infty}\ell_n(x,\mathbf{A})=\infty\right\}\] has Hausdorff dimension at least $1-\tau$.