Calculate epsilon_q

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 $\beta$}