From 15a36cb60e1ff280e28da2eb77d9d19917517a73 Mon Sep 17 00:00:00 2001
From: Terence Tao <tao@math.ucla.edu>
Date: Mon, 5 Aug 2024 16:26:39 -0700
Subject: [PATCH] trying to add graphics to blueprint

---
 blueprint/src/chapter/additive_energy.tex | 4 ++--
 blueprint/src/chapter/beta.tex            | 5 ++---
 blueprint/src/web.tex                     | 2 +-
 3 files changed, 5 insertions(+), 6 deletions(-)

diff --git a/blueprint/src/chapter/additive_energy.tex b/blueprint/src/chapter/additive_energy.tex
index f3dd89a..c7c8fda 100644
--- a/blueprint/src/chapter/additive_energy.tex
+++ b/blueprint/src/chapter/additive_energy.tex
@@ -2,14 +2,14 @@ \chapter{Large value additive energy}
 
 \section{Additive energy}
 
-\begin{definition}[Additive energy]\label{add-def}  Let $W$ be a finite set of real numbers.  The \emph{additive energy} $E_1(W)$ of such a set is defined to be the number of quadruples $(t_1,t_2,t_3,t_4) \in W$ such that
+\begin{definition}[Additive energy]\label{energy-def}  Let $W$ be a finite set of real numbers.  The \emph{additive energy} $E_1(W)$ of such a set is defined to be the number of quadruples $(t_1,t_2,t_3,t_4) \in W$ such that
 $$
 |t_1 + t_2 - t_3 - t_4| \leq 1.$$
 \end{definition}
 
 We remark that in additive combinatorics, the variant $E_0(W)$ of the additive energy is often studied, in which $t_1+t_2-t_3-t_4$ is not merely required to be $1$-bounded, but in fact vanish exactly.  However, this version of additive energy is less relevant for analytic number theory applications.
 
-\begin{lemma}[Basic properties of additive energy]\label{add-energy}\uses{add-def}
+\begin{lemma}[Basic properties of additive energy]\label{add-energy}\uses{energy-def}
 \begin{itemize}
 \item[(i)] If $W$ is a finite set of reals, then
 $$ E_1(W) \asymp \int_\R |\# \{ (t_1,t_2) \in W: |t_1+t_2 - x| \leq 1\} |^2\ dx.$$
diff --git a/blueprint/src/chapter/beta.tex b/blueprint/src/chapter/beta.tex
index b3adf3c..39f7d95 100644
--- a/blueprint/src/chapter/beta.tex
+++ b/blueprint/src/chapter/beta.tex
@@ -10,8 +10,7 @@ \section{Phase functions}
 for all (variable) $u \in [1,2]$ and all fixed $p \geq 0$, where $F^{(p+1)}$ denotes the $(p+1)^{\mathrm{st}}$ derivative of $F$.
 \end{definition}
 
-For instance, $u \mapsto \log u$ is a model phase function (with $\sigma=1$), and for any fixed $\sigma \neq 1$, $u \mapsto u^{1-\sigma}/(1-\sigma)$ is a model phase function.  Informally, a model phase function is a function which asymptotically behaves like
-$u \mapsto \log u$ (for $\sigma = 1$) or $u \mapsto u^{1-\sigma}/(1-\sigma)$ (for $\sigma \neq 1$), up to constants.
+For instance, $u \mapsto \log u$ is a model phase function (with $\sigma=1$), and for any fixed $\sigma \neq 1$, $u \mapsto u^{1-\sigma}/(1-\sigma)$ is also a model phase function.  Informally, a model phase function is a function which asymptotically behaves like $u \mapsto \log u$ (for $\sigma = 1$) or $u \mapsto u^{1-\sigma}/(1-\sigma)$ (for $\sigma \neq 1$), up to constants.  This turns out to be a good class for exponential sum estimates, as it is stable under Weyl differencing and Legendre transforms, which show up in the van der Corput A-process and B-process respectively.
 
 Note from Proposition \ref{auto} that the $o(1)$ decay rate in \eqref{fpu} can be made uniform, after passing to a subsequence if necessary.
 
@@ -40,7 +39,7 @@ \section{Exponential sum exponent}
 \item[(i)] $\beta(\alpha) \leq \overline{\beta}$.
 \item[(ii)] For every (fixed) $\eps>0$ and $\sigma > 0$ there exists (fixed) $\delta>0$, $P \geq 1$, $C \geq 1$ with the following property: if $T \geq C$, $T^{\alpha-\delta} \leq N \leq T^{\alpha+\delta}$ are (fixed) real numbers, $I \subset [N,2N]$ is a (fixed) interval, and $F$ is a (fixed) phase function such that
 \begin{equation}\label{fpu-bound}
-     |F^{(p+1)}(u) - \frac{d^p}{du^p} u^{-\sigma}| \leq \delta
+     \left|F^{(p+1)}(u) - \frac{d^p}{du^p} u^{-\sigma}\right| \leq \delta
 \end{equation}
 for all (fixed) $0 \leq p \leq P$ and $u \in [1,2]$, then
 $$ |\sum_{n \in I} e(T F(n/N))| \leq C T^{\overline{\beta}+\eps}.$$
diff --git a/blueprint/src/web.tex b/blueprint/src/web.tex
index 9d7ac78..60b66b1 100644
--- a/blueprint/src/web.tex
+++ b/blueprint/src/web.tex
@@ -9,7 +9,7 @@
 \usepackage{amssymb, amsthm, amsmath, mathtools}
 \usepackage{hyperref}
 \usepackage[showmore, dep_graph]{blueprint}
-
+\usepackage{graphicx}
 \usepackage{listings}
 \usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math}
 \usepackage[nameinlink, capitalize]{cleveref}