论文标题
最大运行长度函数具有约束：ERDőS-Rényi限制定理的概括和特殊集合
Maximal run-length function with constraints: a generalization of the Erdős-Rényi limit theorem and the exceptional sets
论文作者
论文摘要
令$ \ mathbf {a} = \ {a_i \} _ {i = 1}^{\ infty} $是一组序列，每个$ a_i $都是$ 0 $ - $ 1 $ - $ 1 $ $ i $的非空收藏。对于$ x \在[0,1）$中，最大运行长度函数$ \ ell_n（x，x，\ m马理bf {a}）$（相对于$ \ mathbf {a} $）定义为最大的$ k $，因此在第一个$ x $ $ x $的第一个$ n $ n $ n $ n $ n $ n $ n $ n $ x $中的$ x $ $ x $ a__ $ x $ a__ $ x $ a__ $ x $ a $ a__是连续的$ a__假设$ \ lim_ {n \ to \ infty}（\ log_2 | a_n |）/n =τ$对于某些$τ\ [0,1] $中的某些$τ\，一个额外的假设，我们证明了Erdős-rényi限制的概括，该限制据列出了该限制的限制。 \ [\ lim_ {n \ to \ infty} \ frac {\ ell_n（x，x，x，\ mathbf {a}）} {\ log_2n} = \ frac {1} {1} {1-τ} \]对于lebesgue几乎全部$ x \ in [0,11）$。对于特殊集，我们在$ \ mathbf {a} $上的某个更强的假设下证明了\ [\ left \ {x \ in [0,1）：\ lim_ {n \ to \ infty} \ to \ infty} \ frac \ frac { \ lim_ {n \ to \ infty} \ ell_n（x，x，\ mathbf {a}）= \ infty \ right \} \]的Hausdorff dimension至少$ 1-τ$。
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 $\lim_{n\to\infty}(\log_2|A_n|)/n=τ$ for some $τ\in [0,1]$ and one additional assumption holds, we prove a generalization of the Erdős-Rényi limit theorem which states that \[\lim_{n\to\infty}\frac{\ell_n(x,\mathbf{A})}{\log_2n}=\frac{1}{1-τ}\] 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 $1-τ$.
