From 34f6f7730ad49bade2d74314e5d8c5204af03be0 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sun, 14 May 2023 20:38:38 +0100
Subject: [PATCH] Complete epsilon lemma statement
---
main.tex | 23 +++++++++++++++++++++--
1 file changed, 21 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index 231d55c..31376b6 100644
--- a/main.tex
+++ b/main.tex
@@ -991,9 +991,9 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
\begin{lemmadfn}[
Finding better alternatives to $\epsilon_F$:
- $\epsilon_q^1$ and $\epsilon_q^1$
+ $\epsilon_q^1$ and $\epsilon_q^2$
]
-Suppose $d \in \frac{1}{m}\ZZ$ is satisfies the condition in
+Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
eqn \ref{eqn:positive_rad_condition_in_terms_of_q_beta}.
That is:
@@ -1010,6 +1010,25 @@ Then we have:
\end{equation*}
Where $\epsilon_q^1$ and $\epsilon_q^2$ are defined as follows:
+
+\begin{equation*}
+ \epsilon_q^1 :=
+ \frac{k_q^1}{2mn^2}
+ \qquad
+ \epsilon_q^2 :=
+ \frac{k_q^2}{2mn^2}
+\end{equation*}
+\begin{align*}
+ \text{where }
+ &k_q^1 \text{ is the least }
+ k\in\ZZ_{>0}\: s.t.:\:
+ k \equiv -\aa\bb m \mod n
+\\
+ &k_q^2 \text{ is the least }
+ k\in\ZZ_{>0}\: s.t.:\:
+ k \equiv \aa\bb m (\aa\aa^{'}-2)
+ \mod n\gcd(2n,\aa^2 m)
+\end{align*}
\end{lemmadfn}
--
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