From 1f34aa395c7759348ed4f00eb6fa5e628ed97f7a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Mar=C3=ADa=20In=C3=A9s=20de=20Frutos-Fern=C3=A1ndez?=
 <88536493+mariainesdff@users.noreply.github.com>
Date: Wed, 14 Aug 2024 15:11:52 +0200
Subject: [PATCH] Fix constants in blueprint (#110)

Fix some constants in Lemma 6.1.3.
---
 blueprint/src/chapter/main.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/blueprint/src/chapter/main.tex b/blueprint/src/chapter/main.tex
index 92c6158b..0b2e8d3e 100644
--- a/blueprint/src/chapter/main.tex
+++ b/blueprint/src/chapter/main.tex
@@ -3337,7 +3337,7 @@ \section{The density arguments}\label{sec-TT*-T*T}
 We have that
 \begin{equation}\label{eqttt9}
   \left|\int \overline{g(x)} \sum_{\fp \in \mathfrak{A}} T_{\fp} f(x)\, d\mu(x)\right|\le
-  2^{111a^2}({q}-1)^{-1} \dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} \|f\|_2\|g\|_2\, .
+  2^{111a^3}({q}-1)^{-1} \dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12} \|f\|_2\|g\|_2\, .
 \end{equation}
 \end{lemma}
 \begin{proof}
@@ -3363,7 +3363,7 @@ \section{The density arguments}\label{sec-TT*-T*T}
 \end{equation}
 We have with \Cref{Hardy-Littlewood}
 \begin{equation}
-\left\|M_{\mathcal{B}, \frac {2q}{3q-2}} f\right\|_2\le 2^{2a}(3\tilde{q}-2)(2\tilde{q}-2)^{-1}\|f\|_2\, .
+\left\|M_{\mathcal{B}, \frac {2{\tilde{q}}}{3{\tilde{q}}-2}} f\right\|_2\le 2^{2a}(3\tilde{q}-2)(2\tilde{q}-2)^{-1}\|f\|_2\, .
 \end{equation}
 Using $1<\tilde{q}\le 2$ estimates the last display by
 \begin{equation}\label{eqttt2}
@@ -3378,12 +3378,12 @@ \section{The density arguments}\label{sec-TT*-T*T}
      \le \|g\|_2 \Big\| \sum_{\fp \in \mathfrak{A}} T_{\fp} f \Big\|_2
 \end{equation}
  \begin{equation}
-     \le 2^{107a^2}\|g\|_2 \| M_{\mathcal{B}}f \|_2
+     \le 2^{107a^3}\|g\|_2 \| M_{\mathcal{B}}f \|_2
 \end{equation}
 With \eqref{eqttt1} and
 \eqref{eqttt2} we can estimate the last display by
 \begin{equation}
-    \le 2^{107a^2+2a+2}(\tilde{q}-1)^{-1} \|g\|_2 \|f\|_2\dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12}
+    \le 2^{107a^3+2a+2}(\tilde{q}-1)^{-1} \|g\|_2 \|f\|_2\dens_2(\mathfrak{A})^{\frac 1{\tilde{q}}-\frac 12}
 \end{equation}
 Using $a\ge 4$ and
 $(\tilde q - 1)^{-1} = (q+1)/(q-1) \le 3(q-1)^{-1}$