From 49a9eedb6198faa9eca576e3e9804e7b8cf94a74 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Sat, 13 May 2023 00:37:11 +0100
Subject: [PATCH] Calculate epsilon_q
---
main.tex | 35 +++++++++++++++++++++++++++++++++++
1 file changed, 35 insertions(+)
diff --git a/main.tex b/main.tex
index c01c5fa..a0c4921 100644
--- a/main.tex
+++ b/main.tex
@@ -994,6 +994,41 @@ Considering the numerator of the right-hand-side of
\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:
+
+\begin{align}
+ &\frac{ x }{ m }
+ - \frac{
+ (\aa r+2\bb)\aa
+ }{
+ 2n^2
+ }
+ = \frac{ k }{ 2mn^2 }
+ \quad \text{for some } x \in \ZZ
+\\ &\iff
+ - (\aa r+2\bb)\aa m
+ \equiv k
+ \quad \mod 2n^2
+\\ &\iff
+ - \aa\bb m
+ \equiv k
+ \quad \mod 2n^2
+\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 $q$.
+
+%% TODO: check this^ result seems a bit strange
+
\minorheading{Irrational $\beta$}
--
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