Limitations of deducing measures of limsup sets from measures of finite intersections

TL;DR

This paper demonstrates that the measure bounds for limsup sets established for sequences of balls do not extend to more general sets, highlighting the necessity of the ball condition in such measure-theoretic results.

Contribution

It shows that the condition requiring sets to be balls is essential for measure equivalences in limsup set theorems, providing counterexamples when this condition is dropped.

Findings

The measure of limsup sets can differ significantly when sets are not balls.

Counterexamples exist where sets with identical measures and intersections have different limsup measures.

The necessity of the ball condition is confirmed for measure-theoretic limsup set results.

Abstract

Early results by Borel and Cantelli and Erd\H{o}s 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 $\mu$. 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 $\mu(A_i)=\mu(B_i)$ for all $i \in \mathbb{N}$ and for any sequence $(i_1,i_2,...,i_l)$ where $l \leq m$ we have $\mu(A_{i_1}\cap A_{i_2} \cap... \cap A_{i_l})=\mu(B_{i_1}\cap B_{i_2} \cap... \cap B_{i_l})$ but $\mu(\limsup_{i \rightarrow \infty} A_i)=1$ and $\mu(\limsup_{i \rightarrow \infty} B_i)=0$.