Start leading onto part with tighter bounds

main.tex

@@ -23,7 +23,7 @@
 \newcommand{\centralcharge}{\mathcal{Z}}
 \newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
 
-\newtheorem{rmax_with_uniform_eps}{Theorem}[section]
+\newtheorem{theorem}{Theorem}[section]
 
 \begin{document}
 
@@ -787,6 +787,13 @@ Some of the details around the associated numerics are explored next.
 
 The strategy here is similar to what was shown in (sect \ref{sec:twisted-chern}),
 % ref to Schmidt?
+
+\begin{sagesilent}
+var("a_F b_q n") # Define symbols introduce for values of beta and q
+beta_value_expr = (beta == a_F/n)
+q_value_expr = (q == b_q/n)
+\end{sagesilent}
+
 \renewcommand{\aa}{{a_F}}
 \newcommand{\bb}{{b_q}}
 Suppose $\beta = \frac{\aa}{n}$ for some coprime $n \in \NN,\aa \in \ZZ$.
@@ -799,23 +806,6 @@ Then fix a value of $q$:
 	\cap [0, \chern_1^{\beta}(F)]
 \end{equation}
 as noted at the beginning of this section (\ref{sec:refinement}).
-Firstly, we only consider $r$-values for which $c:=\chern_1(E)$ is integral:
-
-\begin{sagesilent}
-var("a_F b_q n") # Define symbols introduce for values of beta and q
-beta_value_expr = (beta == a_F/n)
-q_value_expr = (q == b_q/n)
-\end{sagesilent}
-
-\begin{equation}
-	c =
-	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
-	\in \ZZ
-\end{equation}
-
-\noindent
-That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
-$n$, and so invertible mod $n$).
 
 Substituting the current values of $q$ and $\beta$ into the condition for the
 radius of the pseudo-wall being positive
@@ -832,7 +822,8 @@ radius of the pseudo-wall being positive
 	\frac{1}{2n^2}\ZZ
 \end{equation}
 
-\begin{rmax_with_uniform_eps}[Bound on $r$ \#1]
+\begin{theorem}[Bound on $r$ \#1]
+\label{thm:rmax_with_uniform_eps}
 	Let $v = (R,C,D)$ be a fixed Chern character. Then the ranks of the
 	pseudo-semistabilizers for $v$ are bounded above by the following expression.
 
@@ -853,7 +844,7 @@ radius of the pseudo-wall being positive
 			\right)
 		\right\}
 	\end{align*}
-\end{rmax_with_uniform_eps}
+\end{theorem}
 
 \begin{proof}
 
@@ -945,8 +936,41 @@ for $\epsilon$ gives the result.
 
 \end{proof}
 
+%% TODO simplified expression for rmax by predicting which q gives rmax
+
 %% refinements using specific values of q and beta
 
+This bound can be refined a bit more by considering restrictions from the
+possible values that $r$ take.
+Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
+that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
+$\frac{1}{m}\ZZ$ is at least $\frac{1}{\lcm(m,2n^2)}$ away. However this a
+conservative estimate, and a larger gap can sometimes be guaranteed if we know
+this value of $\frac{1}{2n^2}\ZZ$ explicitly. 
+
+The expressions that will follow will be a bit more complicated and have more
+parts which depend on the values of $q$ and $\beta$, even their numerators
+$\aa,\bb$ specifically. The upcoming theorem (TODO ref) is less useful as a
+`nice' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
+purpose in the context of writing a computer program to find
+pseudo-semistabilizers. Such a program would iterate through possible values of
+$q$, then iterate through values of $r$ within the bounds (dependent on $q$),
+which would then determine $c$, and then find the corresponding possible values
+for $d$.
+
+
+Firstly, we only consider $r$-values for which $c:=\chern_1(E)$ is integral:
+
+\begin{equation}
+	c =
+	\sage{c_in_terms_of_q.subs([q_value_expr,beta_value_expr])}
+	\in \ZZ
+\end{equation}
+
+\noindent
+That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
+$n$, and so invertible mod $n$).
+
 \begin{sagesilent}
 rhs_numerator = (
 	positive_radius_condition