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Add \Cref to proposition and sub-sub-section (#15)
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pitmonticone authored May 12, 2024
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22 changes: 8 additions & 14 deletions blueprint/src/chapter/main.tex
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Expand Up @@ -271,21 +271,15 @@ \chapter{Proof of Metric Space Carleson, overview}
$\sigma_1$ and $\sigma_2$ are contained in $[-S,S]$ and $F$ and $G$ are contained
in the ball $B(o, D^S)$.

In \Cref{christsection}, we prove \Cref{finitary-Carleson} using a
bound for a dyadic model formulated in \Cref{discrete-Carleson} below.

In \Cref{christsection},
we prove \Cref{finitary-Carleson}
using a
bound for a dyadic model formulated in Proposition
\ref{discrete-Carleson} below.

A grid structure $(\mathcal{D}, c, s)$ on $X$ consists of a finite collection $\mathcal{D}$ of Borel sets in $X$ called dyadic cubes, a surjective function $s\colon \mathcal{D}\to [-S, S]$
called scale function,
and a function $c:\mathcal{D}\to X$
A grid structure $(\mathcal{D}, c, s)$ on $X$ consists of a finite collection $\mathcal{D}$ of Borel
sets in $X$ called dyadic cubes, a surjective function $s\colon \mathcal{D}\to [-S, S]$
called scale function, and a function $c:\mathcal{D}\to X$
called center function such that the five properties
\eqref{coverdyadic},
\eqref{dyadicproperty}, \eqref{coverball},
\eqref{eq-vol-sp-cube}, and \eqref{eq-small-boundary}
hold.
\eqref{coverdyadic}, \eqref{dyadicproperty}, \eqref{coverball},
\eqref{eq-vol-sp-cube}, and \eqref{eq-small-boundary} hold.

For each dyadic cube $I$ and each $-S\le k<s(I)$ we have
\begin{equation}\label{coverdyadic}
Expand Down Expand Up @@ -4801,7 +4795,7 @@ \section{Almost orthogonality of separated trees}
\section{Proof of the Tiles with large separation Lemma}
\label{subsec-big-tiles}

\Cref{correlation-distant-tree-parts} follows from the van der Corput estimate in \Cref{Holder-van-der-Corput}. We apply this proposition in Subsubsection \ref{subsubsec-van-der-corput}. To prepare this application, we first, in Subsubsection \ref{subsubsec-pao}, construct a suitable partition of unity, and show then, in Subsubsection \ref{subsubsec-holder-estimates} the H\"older estimates needed to apply \Cref{Holder-van-der-Corput}.
\Cref{correlation-distant-tree-parts} follows from the van der Corput estimate in \Cref{Holder-van-der-Corput}. We apply this proposition in \Cref{subsubsec-van-der-corput}. To prepare this application, we first, in \Cref{subsubsec-pao}, construct a suitable partition of unity, and show then, in \Cref{subsubsec-holder-estimates} the H\"older estimates needed to apply \Cref{Holder-van-der-Corput}.

\subsection{A partition of unity}
\label{subsubsec-pao}
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