Correct recent work related to modular arithmetic

main.tex

@@ -951,7 +951,7 @@ 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
+`clean' 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 $q$),
@@ -959,7 +959,8 @@ 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:
+Firstly, we only need to consider $r$-values for which $c:=\chern_1(E)$ is
+integral:
 
 \begin{equation}
 	c =
@@ -982,24 +983,9 @@ rhs_numerator = (
 \end{sagesilent}
 
 \noindent
-Considering the numerator of the right-hand-side of
-(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}):
+Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
 
-\begin{align}
-	\sage{rhs_numerator}
-	&\equiv (\aa(-\aa^{-1}\bb)+2\bb)\aa &\mod n
-\\
-	&\equiv \aa\bb &\mod n
-\end{align}
-
-\noindent
-And so, we also have $\aa(\aa r+2\bb) \equiv \aa\bb$ (mod $2n^2$).
-Now, suppose that $x/m$ is the smallest element of $\frac{1}{m}\ZZ$ strictly
-greater than the right-hand-side of
-(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}), and define $\epsilon$
-as the size of the gap.
-
-Using the following tautology:
+Considering the following tautology:
 
 \begin{align}
 	&\frac{ x }{ m }
@@ -1009,25 +995,48 @@ Using the following tautology:
 		2n^2
 	}
 	= \frac{ k }{ 2mn^2 }
-	\quad \text{for some } x \in \ZZ
+	\quad \text{for some } x \in \ZZ, k \in \ZZ_{>0}
+	\label{eqn:finding_better_eps_problem}
 \\ &\iff
 	- (\aa r+2\bb)\aa m
 	\equiv k
 	\quad \mod 2n^2
+	\quad \text{for some } k \in \ZZ_{>0}
 \\ &\iff
-	- \aa\bb m
+	- \aa^2 m r - 2\aa\bb m
 	\equiv k
 	\quad \mod 2n^2
+	\quad \text{for some } k \in \ZZ_{>0}
+\\ &\Longrightarrow
+  \aa^2 \aa^{'}\bb m - 2\aa\bb m
+	\equiv k
+	\quad \mod \gcd(2n^2, \aa^2 mn)
+	\quad \text{for some } k \in \ZZ_{>0}
+	\label{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
+\\ &\Longrightarrow
+  -\aa\bb m
+	\equiv k
+	\quad \mod n
+	\quad \text{for some } k \in \ZZ_{>0}
+	\label{eqn:better_eps_problem_k_mod_n}
 \end{align}
 
-We can recover how much greater $x/m$ is than the right-hand-side of
-(eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}).
-First calculate the smallest $k_q \in \ZZ_{>0}$, such that
-$k_q \equiv -\aa\bb m \mod 2n^2$. Then we have
-$\epsilon = \epsilon_q := \frac{k_q}{2mn^2}$,
-an expression independent of $x$ and $r$, only depending on $q$.
+In our situation, we want to find the gap between the right-hand-side of eqn
+\ref{eqn:positive_rad_condition_in_terms_of_q_beta},
+and the least element of $\frac{1}{m}\ZZ$ which is strictly greater.
+This amounts to finding the least $k \in \ZZ_{>0}$ for which 
+eqn \ref{eqn:finding_better_eps_problem} holds.
+Since such a $k$ satisfies eqn \ref{eqn:better_eps_problem_k_mod_n},
+we can pick the smallest $k \in \ZZ_{>0}$ which satisfies this new condition
+(a computation only depending on $q$ and $\beta$, but not $r$)
+and we are then guaranteed that the gap is at least $\frac{k}{2mn^2}$.
+
+A potentially larger gap can also be guaranteed if we choose the least
+$k \in \ZZ_{>0}$ satisfying eqn \ref{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
+instead, but at the cost of computing several $\gcd$'s and modulo reductions
+for each $q$ considered.
+
 
-%% TODO: check this^ result seems a bit strange