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fpvandoorn committed May 2, 2024
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3 changes: 1 addition & 2 deletions blueprint/src/chapter/main.tex
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Expand Up @@ -1693,8 +1693,7 @@ \section{Proof of Lemma \ref{tile-structure}, tile structure}

1. There exist $z \in \mathcal{Z}(J) \cap \Omega_1(\fp)$ such that $\mfa \in \Omega(J, z)$, and there exists $z' \in \mathcal{Z}(J) \cap \Omega_1(\fp')$ such that $\mfa \in \Omega(J, z')$. By the induction hypothesis, that \eqref{eq-dis-freq-cover} holds for $J$, we must have $z = z'$. By Lemma \ref{disjoint-frequency-cubes}, we must then have $\fp = \fp'$.

2. There exists $z \in \mathcal{Z}(J) \cap \Omega_1(\fp)$ such that $\mfa \in \Omega(J,z)$, and $\mfa \in B_{\fp'}(\fcc(\fp'), 0.2)$. Using the triangle inequality, Lemma \ref{monotone
cube metrics} and \eqref{eq-freq-comp-ball}, we obtain
2. There exists $z \in \mathcal{Z}(J) \cap \Omega_1(\fp)$ such that $\mfa \in \Omega(J,z)$, and $\mfa \in B_{\fp'}(\fcc(\fp'), 0.2)$. Using the triangle inequality, Lemma \ref{monotone-cube-metrics} and \eqref{eq-freq-comp-ball}, we obtain
$$
d_{\fp'}(\fcc(\fp'),z) \le d_{\fp'}(\fcc(\fp'), \mfa) + d_{\fp'}(z, \mfa) \le 0.2 + 2^{-95a} \cdot 1 < 0.3\,.
$$
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