Replace primes in blueprint to dodge bug

blueprint/src/chapter/improved_exponent.tex

@@ -13,7 +13,7 @@ \chapter{Improving the exponents}
 
 We have the following variant of Lemma \ref{construct-good-prelim}:
 
-\begin{lemma}[Constructing good variables, I']\label{construct-good-prelim'}\lean{construct_good_prelim'}\leanok
+\begin{lemma}[Constructing good variables, I']\label{construct-good-prelim-improv}\lean{construct_good_prelim'}\leanok
   One has
   \begin{align*}  k \leq
     \delta + \eta (& d[X^0_1;T_1|T_3]-d[X^0_1;X_1])
@@ -51,15 +51,15 @@ \chapter{Improving the exponents}
 
 (One could in fact refactor Lemma \ref{construct-good-prelim} to follow from Lemma \ref{construct-good-prelim'} and Lemma \ref{cond-dist-fact}).
 
-\begin{lemma}[Constructing good variables, II']\label{construct-good'}\lean{construct_good'}
+\begin{lemma}[Constructing good variables, II']\label{construct-good-improv}\lean{construct_good'}
 One has
 \begin{align*}  k & \leq  \delta + \frac{\eta}{6}  \sum_{i=1}^2 \sum_{1 \leq j,l \leq 3; j \neq l} (d[X^0_i;T_j|T_l] - d[X^0_i; X_i])
   \end{align*}
 \end{lemma}
 
 \begin{proof}
-\uses{construct-good-prelim'}
-Average Lemma \ref{construct-good-prelim'} over all six permutations of $T_1,T_2,T_3$.
+\uses{construct-good-prelim-improv}
+Average Lemma \ref{construct-good-prelim-improv} over all six permutations of $T_1,T_2,T_3$.
 \end{proof}
 
 Now let $X_1, X_2, \tilde X_1, \tilde X_2$ be independent copies of $X_1, X_2, X_1, X_2$, and set
@@ -77,7 +77,7 @@ \chapter{Improving the exponents}
     \end{align*}
 \end{lemma}
 
-\begin{proof}\uses{construct-good', key-ident}  For each $s$ in the range of $S$, apply Lemma \ref{construct-good'} with $T_1,T_2,T_3$ equal to $(U|S=s)$, $(V|S=s)$, $(W|S=s)$ respectively (which works thanks to Lemma \ref{key-ident}), multiply by $\bbP[S=s]$, and sum in $s$ to conclude.
+\begin{proof}\uses{construct-good-improv, key-ident}  For each $s$ in the range of $S$, apply Lemma \ref{construct-good-improv} with $T_1,T_2,T_3$ equal to $(U|S=s)$, $(V|S=s)$, $(W|S=s)$ respectively (which works thanks to Lemma \ref{key-ident}), multiply by $\bbP[S=s]$, and sum in $s$ to conclude.
 \end{proof}
 
 To control the expressions in the right-hand side of this lemma we need a general entropy inequality.
@@ -149,7 +149,7 @@ \chapter{Improving the exponents}
 (actually this is in the proof of that lemma, so a little refactoring is needed here).  Summing the previous two estimates, we obtain the claim.
 \end{proof}
 
-\begin{theorem}[Improved $\tau$-decrement]\label{de-prop'}\lean{tau_strictly_decreases'}\leanok
+\begin{theorem}[Improved $\tau$-decrement]\label{de-prop-improv}\lean{tau_strictly_decreases'}\leanok
 Suppose $0 < \eta < 1/8$.  Let $X_1, X_2$ be tau-minimizers.  Then $d[X_1;X_2] = 0$.
 \end{theorem}
 
@@ -158,7 +158,7 @@ \chapter{Improving the exponents}
   For any $\eta < 1/8$, we see from Lemma \ref{first-estimate} that the expression $\frac{(1 -5 \eta - \frac{4}{6} \eta)(2 \eta k - I_1)}{(1-\eta)}$ is nonnegative, and hence $k = 0$ as required.
 \end{proof}
 
-\begin{theorem}[Improved entropy version of PFR]\label{entropy-pfr'}\lean{entropic_PFR_conjecture_improv}\leanok
+\begin{theorem}[Improved entropy version of PFR]\label{entropy-pfr-improv}\lean{entropic_PFR_conjecture_improv}\leanok
   Let $G = \F_2^n$, and suppose that $X^0_1, X^0_2$ are $G$-valued random variables. Let $0 < \eta < 1/8$.
   Then there is some subgroup $H \leq G$ such that
   \[
@@ -168,16 +168,16 @@ \chapter{Improving the exponents}
   Furthermore, both $d[X^0_1;U_H]$ and $d[X^0_2;U_H]$ are at most $6 d[X^0_1;X^0_2]$.
 \end{theorem}
 
-\begin{proof} \uses{de-prop', tau-min, lem:100pc, ruzsa-triangle}   Let $X_1, X_2$ be the $\tau$-minimizer from Lemma \ref{tau-min}.  From Theorem \ref{de-prop'}, $d[X_1;X_2]=0$.  From Corollary \ref{lem:100pc}, $d[X_1;U_H] = d[X_2; U_H] = 0$.  Also from $\tau$-minimization we have $\tau[X_1;X_2] \leq \tau[X^0_1;X^0_2]$.  Using this and the Ruzsa triangle inequality we can conclude.
+\begin{proof} \uses{de-prop-improv, tau-min, lem:100pc, ruzsa-triangle}   Let $X_1, X_2$ be the $\tau$-minimizer from Lemma \ref{tau-min}.  From Theorem \ref{de-prop-improv}, $d[X_1;X_2]=0$.  From Corollary \ref{lem:100pc}, $d[X_1;U_H] = d[X_2; U_H] = 0$.  Also from $\tau$-minimization we have $\tau[X_1;X_2] \leq \tau[X^0_1;X^0_2]$.  Using this and the Ruzsa triangle inequality we can conclude.
 \end{proof}
 
 One can then replace Theorem \ref{pfr} with
 
-\begin{theorem}[Improved PFR]\label{pfr'}\lean{PFR_conjecture_improv}\leanok
+\begin{theorem}[Improved PFR]\label{pfr-improv}\lean{PFR_conjecture_improv}\leanok
   If $A \subset {\bf F}_2^n$ is non-empty and $|A+A| \leq K|A|$, then $A$ can be covered by most $2K^{11}$ translates of a subspace $H$ of ${\bf F}_2^n$ with $|H| \leq |A|$.
 \end{theorem}
 
-\begin{proof}\uses{entropy-pfr', unif-exist, uniform-entropy-II, jensen-bound,ruz-dist-def,ruzsa-diff,bound-conc,ruz-cov} By repeating the proof of Theorem \ref{pfr} and using Theorem \ref{entropy-pfr'} one can obtain the claim with $11$ replaced by $3 +\frac{1}{\eta}$ for any $0 < \eta < 1/8$.  Now send $\eta$ to $1/8$ and use the fact that there are only finitely many possible choices for the subspace $H$ to conclude.
+\begin{proof}\uses{entropy-pfr-improv, unif-exist, uniform-entropy-II, jensen-bound,ruz-dist-def,ruzsa-diff,bound-conc,ruz-cov} By repeating the proof of Theorem \ref{pfr} and using Theorem \ref{entropy-pfr-improv} one can obtain the claim with $11$ replaced by $3 +\frac{1}{\eta}$ for any $0 < \eta < 1/8$.  Now send $\eta$ to $1/8$ and use the fact that there are only finitely many possible choices for the subspace $H$ to conclude.
 \end{proof}
 
-Of course, by replacing Theorem \ref{pfr} with Theorem \ref{pfr'} we may also improve constants in downstream theorems in a straightforward manner.
+Of course, by replacing Theorem \ref{pfr} with Theorem \ref{pfr-improv} we may also improve constants in downstream theorems in a straightforward manner.