Fragility part

- small typos
- def of $l_{i,j}$
- removed $i(H)$, 
- reintroduced the definition of tree decomposition (T,b)

comparable-box-dimension.tex

@@ -708,7 +708,7 @@ by Erlebach et al.~\cite{erlebach2005polynomial}.
 
 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 $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$.
+if for every $x\in B$, there exists a translation $A'$ of $A$ such that $x\in A'$ and $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$.
 Note that for two boxes $A$ and $B$, we have $A\sqsubseteq_1 B$ if and only if $A\sqsubseteq B$.
 An \emph{$s$-comparable envelope representation} $(\iota,\omega)$ of a graph $G$ in $\mathbb{R}^d$ consists of 
 two functions $\iota,\omega:V(G)\to 2^{\mathbb{R}^d}$ such that for some ordering $v_1$, \ldots, $v_n$ of vertices of $G$,
@@ -729,40 +729,71 @@ the smallest axis-aligned hypercube containing $f(v)$, then there exists a posit
 $(f,\omega)$ is an $s_d$-comparable envelope representation of $G$ in $\mathbb{R}^d$ of thickness at most $2$.
 \end{itemize}
 
+\note{TO REMOVE if we reintroduce tree decomposition earlier !}
+
+\note{Let us recall some notions about treewidth.
+  A \emph{tree decomposition} of a graph $G$ is a pair
+  $(T,\beta)$, where $T$ is a rooted tree and $\beta:V(T)\to 2^{V(G)}$
+  assigns a \emph{bag} to each of its nodes, such that
+\begin{itemize}
+\item for each $uv\in E(G)$, there exists $x\in V(T)$ such that
+  $u,v\in\beta(x)$, and
+\item for each $v\in V(G)$, the set $\{x\in V(T):v\in\beta(x)\}$ is
+  non-empty and induces a connected subtree of $T$.
+\end{itemize}
+For nodes $x,y\in V(T)$, we write $x\preceq y$ if $x=y$ or $x$ is a descendant of $y$ in $T$.
+%For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in \beta(x)$ and $x$ is nearest to the root of $T$.
+%The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$,
+%and its
+The \emph{width} of the tree decomposition is the maximum of the sizes of the bags minus $1$.  The \emph{treewidth} of a graph is the minimum
+of the widths of its tree decompositions.}
+
 \begin{theorem}\label{thm-twfrag}
 For positive integers $t$, $s$, and $d$, the class of graphs
 with an $s$-comparable envelope representation in $\mathbb{R}^d$ of thickness at most $t$
-is fractionally treewidth-fragile.
+is fractionally treewidth-fragile, with a function $f(k) = O_{t,s,d}\bigl(k^{d}\bigr)$.
 \end{theorem}
 \begin{proof}
 For a positive integer $k$, let $f(k)=(2ksd+2)^dst$.
 Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$
 in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$.
-Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as follows.
+Let us define $\ell_{i,j}\in \mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $|ksd\omega(v_i)[j]|$ such that $\ell_{i-1,j} / \ell_{i,j}$ is a positive integer. Formally
+it is defined as follows.
 \begin{itemize}
 \item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
-\item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j} / b_{i,j}$, where
-$b_{i,j}=\max\bigl(1,\lfloor \tfrac{\ell_{i-1,j}}{ksd|\omega(v_i)[j]|} \bigr)$.
+\item For $i=2,\ldots, n$, let $\ell_{i,j} = \ell_{i-1,j}$, if
+  $\ell_{i-1,j} < ksd|\omega(v_i)[j]|$, and otherwise let
+  $\ell_{i,j}$ be lowest fraction of $\ell_{i-1,j}$ that is
+  greater than $ksd|\omega(v_i)[j]|$, formally $\ell_{i,j} =
+  \min\{\ell_{i-1,j}/b \ |\ b\in
+  \mathbb{N}^+ \text{ and } \ell_{i-1,j}/b \ge ksd|\omega(v_i)[j]|\}$.
 \end{itemize}
-Let $x_j\in [0,\ell_{1,j}]$ be chosen uniformly at random, and let $\HH_j$ be the set of hyperplanes in $\mathbb{R}^d$
-consisting of the points whose $j$-th coordinate is equal to $x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$ and $i\in\{1,\ldots, n\}$.
-For each hyperplane $H\in \HH_j$, let $i(H)$ denote the minimum integer $i$ such that $H$ can be expressed in this form.
-
-Let $\HH=\bigcup_{j=1}^d \HH_j$.
-We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such that the box $\omega(v_a)$ intersects some hyperplane $H\in \HH$
-such that $i(H)\le a$.  First, let us argue that $\text{Pr}[v_a\in X]\le 1/k$.  Indeed, since $\ell{i,j}$ is an integer multiple
-of $\ell_{a,j}$ for every $i<a$, we conclude that $v\in X$ if and only if for some $j\in\{1,\ldots,d\}$ and $m\in\mathbb{Z}$,
-we have $x_j+m\ell_{a,j}\in \omega(v_a)[j]$.  The set $[0,\ell_{1,j}]\cap \bigcup_{m\in\mathbb{Z}} (\omega(v_a)[j]-m\ell_{a,j})$
+Let the real $x_j\in [0,\ell_{1,j}]$ be chosen uniformly at random,
+and let $\HH^i_j$ be the set of hyperplanes in $\mathbb{R}^d$
+consisting of the points whose $j$-th coordinate is equal to
+$x_j+m\ell_{i,j}$ for some $m\in\mathbb{Z}$. As $\ell_{i,j}$ is a
+multiple of $\ell_{i',j}$ whenever $i\le i'$, we have that $\HH^i_j
+\subseteq \HH^{i'}_j$ whenever $i\le i'$.  For $i\in\{1,\ldots,n\}$,
+the \emph{$i$-grid} is $\HH^i=\bigcup_{j=1}^d \HH^i_j$, and we let the
+$0$-grid $\HH^0=\emptyset$.  Similarly as above we have that $\HH^i
+\subseteq \HH^{i'}$ whenever $i\le i'$.
+
+We let $X\subseteq V(G)$ consist of the vertices $v_a\in V(G)$ such
+that the box $\omega(v_a)$ intersects some hyperplane $H\in \HH^a$,
+that is such that $x_j+m\ell_{a,j}\in \omega(v_a)[j]$, for some
+$j\in\{1,\ldots,d\}$ and some $m\in \mathbb{Z}$.  First, let us argue
+that $\text{Pr}[v_a\in X]\le 1/k$.  Indeed, the set $[0,\ell_{1,j}]\cap \bigcup_{m\in\mathbb{Z}} (\omega(v_a)[j]-m\ell_{a,j})$
 has measure $\tfrac{\ell_{1,j}}{\ell_{a,j}}\cdot |\omega(v_a)[j]|$, implying that for fixed $j$, this happens with probability
-$|\omega(v_a)[j]|/\ell_{a,j}$.  Let $a'\le a$ be the largest integer such that $b_{a',j}\neq 1$ if such an index exists,
+$|\omega(v_a)[j]|/\ell_{a,j}$.  Let $a'$ be the largest integer such
+that $a'\le a$ and $\ell_{a',j} < \ell_{a'-1,j}$ if such an index exists,
 and $a'=1$ otherwise; note that $\ell_{a,j}=\ell_{a',j}\ge ksd|\omega(v_{a'})[j]|$.  Moreover, since
 $\omega(v_a)\sqsubseteq_s\iota(v_{a'})\subseteq \omega(v_{a'})$, we have $\omega(v_a)[j]\le s\omega(v_{a'})[j]$.
 Combining these inequalities,
 $$\frac{|\omega(v_a)[j]|}{\ell_{a,j}}\le \frac{s\omega(v_{a'})[j]}{ksd|\omega(v_{a'})[j]|}=\frac{1}{kd}.$$
 By the union bound, we conclude that $\text{Pr}[v_a\in X]\le 1/k$.
 
-Let us now bound the treewidth of $G-X$.  For $a\in\{1,\ldots,n\}$, the \emph{$a$-grid} is $F_a=\bigcup_{H\in \HH:i(h)\le a} H$, and we let
-the $0$-grid $F_0=\emptyset$.  For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus F_a$.
+Let us now bound the treewidth of $G-X$.  
+For $a\ge 0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus (\cup_{H\in \HH^a} H)$.
 A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge 0$.
 A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.
 Note that there exists a rooted tree $T$ whose vertices are
@@ -783,13 +814,14 @@ Finally, let us bound the width of the decomposition $(T,\beta)$.  Let $C$ be a
 is an $a$-cell.  Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$.  Consider $j\in\{1,\ldots,d\}$:
 \begin{itemize}
 \item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.
-\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $b_{a,j}=1$, implying $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$.
-\item If $a>1$ and $\ell_{a,j}>\ell_{a-1,j}$, then $\ell_{a-1,j}\ge 2ksd|\omega(v_a)[j]|$ and
-$$\ell_{a,j}=\frac{\ell_{a-1,j}}{\lfloor \frac{\ell_{a-1,j}}{ksd|\omega(v_a)[j]|}\rfloor}<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$
+\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$ (otherwise $\ell_{a,j}=\ell_{a-1,j}/b$ for some integer $b\ge 2$).
+\item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integers $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that
+$$\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$
 \end{itemize}
 Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$.  Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
 For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$,
-there exists a translation $B_i\subseteq C'$ of $\omega(v_a)$ such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$.
+there exists a translation $B_i$ of $\omega(v_a)$ that intersects $C\cap \iota(v_i)$ and such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$.
+Note that as $B_i$ intersects $C$, we have that $B_i\subseteq C'$.
 Since the representation has thickness at most $t$,
 \begin{align*}
 \vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\