Limitations of deducing measures of limsup sets from measures of finite intersections
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918135425138688 |
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| author | Wilson, Charlie |
| author_facet | Wilson, Charlie |
| contents | Early results by Borel and Cantelli and Erdős and Chung 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 \cite{Velanipaper} states that, for sequences of balls the measure of the corresponding limsup set being positive is equivalent to a condition on the relationship between 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 this statement. 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 $l \leq m$ 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 $μ(\limsup_{i \rightarrow \infty} B_i)=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02416 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limitations of deducing measures of limsup sets from measures of finite intersections Wilson, Charlie Dynamical Systems Early results by Borel and Cantelli and Erdős and Chung 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 \cite{Velanipaper} states that, for sequences of balls the measure of the corresponding limsup set being positive is equivalent to a condition on the relationship between 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 this statement. 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 $l \leq m$ 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 $μ(\limsup_{i \rightarrow \infty} B_i)=0$. |
| title | Limitations of deducing measures of limsup sets from measures of finite intersections |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2502.02416 |