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

Let $\mathbf{A}=\{A_i{\}}_{i=1}^{\mathrm{\infty }}$ 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 ${\mathrm{\ell }}_n(x,\mathbf{A})$ (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 ${\lim }_{n\to \mathrm{\infty }}({\log }_2\mathrm{\mid }{A}_n\mathrm{\mid })\mathrm{/}n=\tau$ for some $\tau \in [0,1]$ and one additional assumption holds, we prove a generalization of the Erdős-Rényi limit theorem which states that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{{\mathrm{\ell }}_n(x,\mathbf{A})}{{\log }_{2}n}=\frac{1}{1-\tau }$$ 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 $$\{x\in [0,1):\underset{n\to \mathrm{\infty }}{\lim }\frac{{\mathrm{\ell }}_n(x,\mathbf{A})}{{\log }_{2}n}=0\text{ and }\underset{n\to \mathrm{\infty }}{\lim }{\mathrm{\ell }}_n(x,\mathbf{A})=\mathrm{\infty }\}$$ has Hausdorff dimension at least $1-\tau$.