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Commit a05c03d5 authored by Zdenek Dvorak's avatar Zdenek Dvorak
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Added a sentence after the disk graph explanation paragraph.

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...@@ -893,6 +893,7 @@ Then for a $(2k\ell \times 2k\ell)$-cell $C$, $\beta(C)$ contains all the disks ...@@ -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 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 $|\beta(C)| \le 16t(2k+1)^2 / \pi$. 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 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 $A\sqsubseteq_s B$ 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 $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$. 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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