从有限交集的测度推导limsup集的测度的局限性
Limitations of deducing measures of limsup sets from measures of finite intersections
AI总结：
本文指出Beresnevich和Velani的结果中球集条件的必要性，并构造了不满足该条件但limsup集测度不同的例子。
摘要：
早期由Borel和Cantelli以及Chung（Rendiconti del Circolo Matematico di Palermo，1909）和Erdős（Transactions of the AMS，1952）的结果为limsup集的测度提供了以构成集及其交集的测度为条件的界。Beresnevich和Velani（Journal of Mathematical Analysis and its Applications，2023）最近的工作指出，对于球集序列，对应的limsup集测度为正等价于这些球的测度及其 pairwise 交集的测度之间的关系条件。本文证明了Beresnevich和Velani结果中球集条件的严格必要性。此外，令$d \in \mathbb{N}$，令$[0,1]^d$配备Lebesgue测度$μ$。固定$m \in \mathbb{N}$。当去掉集合为球的条件时，我们可以找到两个集合序列$(A_i)_{i \in \mathbb{N}}$和$(B_i)_{i \in \mathbb{N}}$在$[0,1]^d$中，使得对于所有$i \in \mathbb{N}$，有$μ(A_i)=μ(B_i)$且对于任何序列$(i_1,i_2,...,i_l)$，其中对于所有$l \leq m$，有$μ(A_{i_1}\cap A_{i_2} \cap... \cap A_{i_l})=μ(B_{i_1}\cap B_{i_2} \cap... \cap B_{i_l})$但$μ(\limsup_{i \rightarrow \infty} A_i)=1$且$μ(\limsup_{i \rightarrow \infty} B_i)=0$。
英文摘要：
Early results by Borel and Cantelli and Chung (Rendiconti del Circolo Matematico di Palermo, 1909) and Erdős (Transactions of the AMS, 1952) have provided bounds for the measure of a limsup set in terms of measures of its constituent sets and their intersections. Recent work by Beresnevich and Velani (Journal of Mathematical Analysis and its Applications, 2023) states that for sequences of balls the measure of the corresponding limsup set being positive is equivalent to a condition on the relationship between the measures of these balls and their pairwise intersections. In this paper, we show that the condition that the sets are balls is strictly necessary in the result of Beresnevich and Velani. Moreover, let $d \in \mathbb{N}$ and let $[0,1]^d$ be equipped with Lebesgue measure $μ$. Fix $m \in \mathbb{N}$. When we drop the condition that the sets are balls, we can find two sequences of sets $(A_i)_{i \in \mathbb{N}}$ and $(B_i)_{i \in \mathbb{N}}$ in $[0,1]^d$ such that $μ(A_i)=μ(B_i)$ for all $i \in \mathbb{N}$ and for any sequence $(i_1,i_2,...,i_l)$ where for all $l \leq m$ we have $μ(A_{i_1}\cap A_{i_2} \cap... \cap A_{i_l})=μ(B_{i_1}\cap B_{i_2} \cap... \cap B_{i_l})$ but $μ(\limsup_{i \rightarrow \infty} A_i)=1$ and $μ(\limsup_{i \rightarrow \infty} B_i)=0$.
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