From a05c03d555a26f2d0ae81629d3543140e183448e Mon Sep 17 00:00:00 2001
From: Zdenek Dvorak <rakdver@kam.mff.cuni.cz>
Date: Fri, 11 Mar 2022 11:21:54 +0100
Subject: [PATCH] Added a sentence after the disk graph explanation paragraph.

---
 arxiv_cbd.tex | 1 +
 1 file changed, 1 insertion(+)

diff --git a/arxiv_cbd.tex b/arxiv_cbd.tex
index 829c42b..10a9654 100644
--- a/arxiv_cbd.tex
+++ b/arxiv_cbd.tex
@@ -893,6 +893,7 @@ Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks
 To see that such bag is bounded consider the $((2k+1)\ell \times (2k+1)\ell)$ square $C'$ centered on $C$, and note that any
 disk in $\beta(C)$ intersects $C'$ on an area at least $\pi\ell^2/16$. This implies that $|\beta(C)| \le 16t(2k+1)^2 / \pi$.
 
+Let us now give a detailed proof in a more general setting.
 For a measurable set $A\subseteq \mathbb{R}^d$, let $\vol(A)$ denote the Lebesgue measure of $A$.
 For two measurable subsets $A$ and $B$ of $\mathbb{R}^d$ and a positive integer $s$, we write $A\sqsubseteq_s B$
 if for every $x\in B$, there exists a translation $A'$ of $A$ such that $x\in A'$ and $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$.
-- 
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