Merge pull request #1 from teorth/fix-typos

Fix typos

blueprint/src/chapter/additive_energy.tex

@@ -38,7 +38,7 @@ \section{Additive energy}
 
 For (ii), we can upper bound the indicator function of $[-1,1]$ by the Fourier transform of a non-negative bump function $\varphi$, so that the right-hand side is bounded by
 $$ \sum_{t_1,t_2,t_3,t_4 \in W} \varphi(t_1+t_2-t_3-t_4)$$
-which is then bounded by $O(E_1(W))$ by choosing the suppport of $\varphi$ appropriately.  The lower bound is established similarly (using the arguments in (i) to adjust the error tolerance $1$ in the constraint $ |t_1+t_2 - x| \leq 1$ as necessary.)
+which is then bounded by $O(E_1(W))$ by choosing the support of $\varphi$ appropriately.  The lower bound is established similarly (using the arguments in (i) to adjust the error tolerance $1$ in the constraint $ |t_1+t_2 - x| \leq 1$ as necessary.)
 
 For (iii), first observe we may remove duplicates and assume that the $W_i$ are disjoint, then we can use (ii) and the triangle inequality.
 
@@ -73,7 +73,7 @@ \section{Additive energy}
 
 To relate $S(N,W)$ to $E_1(W)$, we first observe the following lemma, implicit in \cite{heath_brown_consecutive_II} and made more explicit in \cite[Lemma 11.4]{guth-maynard}.
 
-\begin{lemma} Let $T \geq 1$. If $a_n$ is a $1$-bounded sequence on $[N,2N]$ for some $1 \leq N \ll T^{O(1)}$, $W$ is $1$-separated in $[-T,T]$, and 
+\begin{lemma} Let $T \geq 1$. If $a_n$ is a $1$-bounded sequence on $[N,2N]$ for some $1 \leq N \ll T^{O(1)}$, $W$ is $1$-separated in $[-T,T]$, and
 $$|\sum_{n \in [N,2N]} a_n n^{-it}| \geq V$$
 for all $t \in W$ and some $V>0$, then
 $$ V^2 E_1(W) \ll T^{o(1)} \sum_{n,m \in [N,2N]} |R_W(n/m)|^3 + T^{-50}.$$
@@ -139,18 +139,18 @@ \section{Large value additive energy region}
 
 Because the cardinality $|W|$ and additive energy $E_1(W)$ of a set $W$ are correlated with each other, as well as with the double zeta sum $S(N,W)$, we will not be able to consider them separately, and instead we will need to consider the possible joint exponents for these two quantities.  We formalize this via the following set:
 
-\begin{definition}[Large value energy region]\label{lv-edef} The \emph{large value energy region} $\Energy \subset \R^5$ is defined to be the set of all fixed tuples $(\sigma,\tau,\rho,\rho^*,s)$ with $1/2 \leq \sigma \leq 1$, $\tau, \rho, \rho' \geq 0$, such that there exists an unbounded $N > 1$, $T = N^{\tau+o(1)}$, $V = N^{\sigma+o(1)}$, a $1$-bounded sequence  $a_n$ on $[N,2N]$, and a  $1$-separated subset $W$ of cardinality $N^{\rho+o(1)}$ in an interval $J$ of length $T$ such that 
+\begin{definition}[Large value energy region]\label{lv-edef} The \emph{large value energy region} $\Energy \subset \R^5$ is defined to be the set of all fixed tuples $(\sigma,\tau,\rho,\rho^*,s)$ with $1/2 \leq \sigma \leq 1$, $\tau, \rho, \rho' \geq 0$, such that there exists an unbounded $N > 1$, $T = N^{\tau+o(1)}$, $V = N^{\sigma+o(1)}$, a $1$-bounded sequence  $a_n$ on $[N,2N]$, and a  $1$-separated subset $W$ of cardinality $N^{\rho+o(1)}$ in an interval $J$ of length $T$ such that
     \begin{equation}\label{sig-large} \left|\sum_{n \in [N,2N]} a_n n^{-it} \right| \geq V
 \end{equation}
 for all $t \in W$, and such that $E_1(W) = N^{\rho^*+o(1)}$ and $S(N,W) = N^{s+o(1)}$.
 
-We define the \emph{large value energy region for zeta} $\Energy_\zeta \subset \R^5$ similarly, but now the interval $J$ is required to be of the form $[T,2T]$, and the sequence $a_n$ is required to be of the form $1_I(n)$ for some interval $I \subset [N,2N]$.  Thus, in order for $(\sigma,\tau,\rho,\rho^*,s)$ to lie in $\Energy_\zeta$, there must exist an unbounded $N > 1$, $T = N^{\tau+o(1)}$, $V = N^{\sigma+o(1)}$, an interval $I$ in $[N,2N]$, and  $W = W$ is a $1$-separated subset of cardinality $N^{\rho+o(1)}$ in $[T,2T]$ such that 
+We define the \emph{large value energy region for zeta} $\Energy_\zeta \subset \R^5$ similarly, but now the interval $J$ is required to be of the form $[T,2T]$, and the sequence $a_n$ is required to be of the form $1_I(n)$ for some interval $I \subset [N,2N]$.  Thus, in order for $(\sigma,\tau,\rho,\rho^*,s)$ to lie in $\Energy_\zeta$, there must exist an unbounded $N > 1$, $T = N^{\tau+o(1)}$, $V = N^{\sigma+o(1)}$, an interval $I$ in $[N,2N]$, and  $W = W$ is a $1$-separated subset of cardinality $N^{\rho+o(1)}$ in $[T,2T]$ such that
 \begin{equation}\label{sig-large-zeta} \left|\sum_{n \in I} n^{-it} \right| \geq V
 \end{equation}
 for all $t \in W$, and such that $E_1(W) = N^{\rho^*+o(1)}$ and $S(N,W) = N^{s+o(1)}$.
 \end{definition}
 
-Clearly we have 
+Clearly we have
 
 \begin{lemma}[Trivial containment] We have $\Energy_\zeta \subset \Energy$.
 \end{lemma}
@@ -193,7 +193,7 @@ \section{Large value additive energy region}
 
 This lemma is proven by a routine expansion of the definitions, and is omitted.
 
-\begin{lemma}[Basic properties]\label{lve-basic}\   
+\begin{lemma}[Basic properties]\label{lve-basic}\
     \begin{itemize}
         \item[(i)] (Monotonicity in $\sigma$) If $(\sigma,\tau,\rho,\rho^*,s) \in \Energy$, then
         $(\sigma',\tau',\rho,\rho^*,s) \in \Energy$ for all $1/2 \leq \sigma' \leq \sigma$ and $\tau' \geq \tau$.
@@ -212,7 +212,7 @@ \section{Large value additive energy region}
 \end{proof}
 
 
-\begin{lemma}[Raising to a power]\label{power-energy}  If $(\sigma,\tau,\rho,\rho^*,s) \in \Energy$, then $(\sigma,\tau/k, \rho/k, (\rho^*)/k,s/k) \in \Energy$ for any integer $k \geq 1$.  
+\begin{lemma}[Raising to a power]\label{power-energy}  If $(\sigma,\tau,\rho,\rho^*,s) \in \Energy$, then $(\sigma,\tau/k, \rho/k, (\rho^*)/k,s/k) \in \Energy$ for any integer $k \geq 1$.
 \end{lemma}
 
 \begin{proof}
@@ -297,7 +297,7 @@ \section{Known relations for the large value energy region}
 for some $0 \leq \kappa \leq \rho$ with
 $$ \kappa + \rho' \leq 2\rho$$
 $$ 2\kappa + \rho' \leq \rho^*$$
-In particular, 
+In particular,
 $$ 2\kappa + 5\rho'/4 \leq 3\rho^*/4 + \rho$$
 and the claim follows after moving the $\kappa$ inside the second maximum and performing some algebra.
 \end{proof}
@@ -341,10 +341,10 @@ \section{Known relations for the large value energy region}
 
 \begin{proof} This follows from \cite[Propositions 4.6, 5.1, 6.1, 8.1, 10.1, (5.5)]{guth-maynard}.
 \end{proof}
- 
+
 \begin{lemma}[Second Guth-Maynard relation]\cite[Lemma 1.7]{guth-maynard}  If $(\sigma,\tau,\rho,\rho^*,s) \in \Energy$ then
 $$ \rho^* \leq \rho + s - 2\sigma.$$
-In particular, from Lemma \ref{hb-double} we see for $\tau \leq 3/2$ that    
+In particular, from Lemma \ref{hb-double} we see for $\tau \leq 3/2$ that
 $$ \rho^* \leq \max(3\rho+1-2\sigma, 2\rho+2-2\sigma).$$
 \end{lemma}
 
@@ -367,7 +367,7 @@ \section{Known relations for the large value energy region}
 
 
 \begin{theorem}[Guth--Maynard large values theorem, again]\label{guth-maynard-lvt-again} One has
-    $$ \LV(\sigma,\tau) \leq \max(2-2\sigma, 18/5 - 4 \sigma, \tau + 12/5 - 4\sigma).$$  
+    $$ \LV(\sigma,\tau) \leq \max(2-2\sigma, 18/5 - 4 \sigma, \tau + 12/5 - 4\sigma).$$
 \end{theorem}
 
 \begin{proof} For $\sigma \leq 7/10$ this follows from Lemma \ref{l2-mvt}, and for $\sigma \geq 8/10$ it follows from Lemma \ref{huxley-lv}.  Thus we may assume that $7/10 \leq \sigma \leq 8/10$.  By subdivision (Lemma \ref{lv-basic}(ii)) it then suffices to treat the case $\tau = 6/5$, that is to say to show that
@@ -387,5 +387,3 @@ \section{Known relations for the large value energy region}
 Inserting this and the $S_2$ bound (with $k=4$) into the bound for $\rho$ and simplifying (using $\tau=6/5$), we eventually obtain
 the desired bound $\rho \leq 18/5-4\sigma$.
 \end{proof}
-
-

blueprint/src/chapter/notation.tex

@@ -40,12 +40,12 @@ \section{Asymptotic (or ``cheap nonstandard'') notation}
 
 \begin{proposition}[Automatic uniformity]\label{auto}  Let $E = E_{\i}$ be a non-empty variable set, and let $f = f_{\i}: E \to \C$ be a variable function.  \begin{itemize}
     \item[(i)] Suppose that $f(x)=O(1)$ for all (variable) $x \in E$.  Then after passing to a subsequence if necessary, the bound is uniform, that is to say, there exists a fixed $C$ such that $|f(x)| \leq C$ for all $x \in E$.
-    \item[(ii)] Suppose that $f(x)=o(1)$ for all (variable) $x \in E$.  Then after passing to a subsequence if necessary, the bound is uniform, that is to say, there exists an infitesimal $c$ such that $|f(x)| \leq c$ for all $x \in E$.
+    \item[(ii)] Suppose that $f(x)=o(1)$ for all (variable) $x \in E$.  Then after passing to a subsequence if necessary, the bound is uniform, that is to say, there exists an infinitesimal $c$ such that $|f(x)| \leq c$ for all $x \in E$.
 \end{itemize}
 \end{proposition}
 
-\begin{proof} We begin with (i).  Suppoe that there is no uniform bound.  Then for any fixed natural number $n$, one can find arbitrarily large $\i_n$ in the sequence and $x_{\i_n} \in E_{\i_n}$ such that $|f_{\i_n}(x_{\i_n})| \geq n$.  Clearly one can arrange matters so that the sequence $\i_n$ is increasing.  If one then restricts to this sequence and sets $x$ to be the variable element $x_{\i_n}$ of $E$, then $f(x)$ is unbounded, a contradiction.
-    
+\begin{proof} We begin with (i).  Suppose that there is no uniform bound.  Then for any fixed natural number $n$, one can find arbitrarily large $\i_n$ in the sequence and $x_{\i_n} \in E_{\i_n}$ such that $|f_{\i_n}(x_{\i_n})| \geq n$.  Clearly one can arrange matters so that the sequence $\i_n$ is increasing.  If one then restricts to this sequence and sets $x$ to be the variable element $x_{\i_n}$ of $E$, then $f(x)$ is unbounded, a contradiction.
+
 Now we prove (ii).  We can assume for each fixed $n$ that there exists $\i_n$ in the ambient sequence such that $|f_{\i}(x_{\i})| \leq 1/n$ for all $\i \geq \i_n$ and $x_{\i} \in E_{\i}$, since if this were not the case one can construct an $x = x_{\i} \in E$ such that $|f_{\i}(x_{\i})| \geq 1/n$ for $\i$ sufficiently large, contradicting the hypothesis.  Again, we may take the $\i_n$ to be increasing.  Restricting to this sequence, and writing $c_{\i_n} := 1/n$, we see that $c=o(1)$ and $|f(x)| \leq c$ for all $x \in E$, as required.
 \end{proof}
 
@@ -54,4 +54,4 @@ \section{Asymptotic (or ``cheap nonstandard'') notation}
 
 
 \begin{remark} There are two caveats to keep in mind when using this asymptotic formalism.  Firstly, the law of the excluded middle is only valid after passing to subsequences.  For instance, it is possible for a nonstandard natural number to neither be even or odd, since it could be even for some $\i$ and odd for others.  However, one can pass to a subsequence in which it becomes either even or odd.  Secondly, one cannot combine the ``external'' concepts of asymptotic notation with the ``internal'' framework of (variable) set theory.  For instance, one cannot view the collection of all bounded (variable) real numbers as a variable set, since the notion of boundedness is not ``pointwise'' to each index $\i$, but instead describes the ``global'' behavior of this index set.  Thus, for instance, set builder notation such as $\{ x: x = O(1) \}$ should be avoided.
-\end{remark}
+\end{remark}

blueprint/src/chapter/primes.tex

@@ -34,7 +34,7 @@ \chapter{Applications to the primes}\label{primes-sec}
 \begin{proof} These are all immediate, after noting from the prime number theorem that $\sum_{p_n \leq x} p_{n+1} - p_n = x^{1+o(1)}$.
 \end{proof}
 
-The Cram\'er random mdoel {\bf give cite} predicts
+The Cram\'er random model {\bf give cite} predicts
 
 \begin{conjecture}[Prime gap conjecture] $\PNTALL = 0$, and hence (by Lemma \ref{pnt-triv}) $\PNTAA = \GAPMAX=0$ and $\GAPSQUARE=1$.
 \end{conjecture}
@@ -43,9 +43,9 @@ \chapter{Applications to the primes}\label{primes-sec}
 
 A basic connection with zero density exponents is
 
-\begin{proposition}[Zero density theorems and prime gaps]\label{prime-gap}  Let 
+\begin{proposition}[Zero density theorems and prime gaps]\label{prime-gap}  Let
 \begin{equation}\label{A-def}
-    \|\A\|_\infty \coloneqq \sup_{1/2 \leq \sigma \leq 1} A(\sigma).  
+    \|\A\|_\infty \coloneqq \sup_{1/2 \leq \sigma \leq 1} A(\sigma).
 \end{equation}
 Then
     $$ \PNTALL \leq 1 - \frac{1}{\|\A\|_\infty}$$
@@ -65,7 +65,7 @@ \chapter{Applications to the primes}\label{primes-sec}
 \begin{corollary}[Ingham-Guth-Maynard bound]\cite{guth-maynard}  We have $\PNTALL \leq \frac{17}{30}$ and $\PNTAA \leq \frac{2}{15}$.
 \end{corollary}
 
-These are currently the best known upper bounds on $\PNTALL$ and $\PNTAA$.  
+These are currently the best known upper bounds on $\PNTALL$ and $\PNTAA$.
 
 \begin{proof}  From Theorem \ref{thm:ingham_zero_density2} and Theorem \ref{guth-maynard-density} one as $\|\A\|_\infty \leq 30/13$, and the claim now follows from Proposition \ref{prime-gap}.
 \end{proof}
@@ -119,7 +119,7 @@ \chapter{Applications to the primes}\label{primes-sec}
 
 This proposition recovers several known bounds (both conditional and unconditional) on $\GAPSQUARE$:
 
-\begin{corollary}\ 
+\begin{corollary}\
     \begin{itemize}
     \item[(i)] Assuming the Riemann hypothesis, $\GAPSQUARE = 1$.  {\bf Selberg}
     \item[(ii)] Unconditionally, one has $\GAPSQUARE \leq 4/3$.  \cite{heath_brown_consecutive_I}