Maximal run-length function with constraints: a generalization of the Erdős–Rényi limit theorem and the exceptional sets

Abstract

Let ${\textbf{A}}=\{A_i\}_{i=1}^{\infty }$ 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 $\ell _n(x,{\textbf{A}})$ (with respect to ${\textbf{A}}$) is defined to be the largest k such that in the first n digits of the dyadic expansion of x there is a consecutive subsequence in $A_k$. Suppose that $\lim _{n\rightarrow \infty }(\log _2|A_n|)/n=\tau$ for some $\tau \in [0,1]$ and one additional assumption holds, we prove a generalization of the Erdős–Rényi limit theorem which states that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\ell _n(x,{\textbf{A}})}{\log _2n}=\frac{1}{1-\tau } \end{aligned}$$

for Lebesgue almost all $x\in [0,1)$. For the exceptional sets, we prove under a certain stronger assumption on ${\textbf{A}}$ that the set

$$\begin{aligned} \left\{ x\in [0,1): \lim _{n\rightarrow \infty }\frac{\ell _n(x,{\textbf{A}})}{\log _2n}=0\;\text {and}\; \lim _{n\rightarrow \infty }\ell _n(x,{\textbf{A}})=\infty \right\} \end{aligned}$$

has Hausdorff dimension at least $1-\tau$.