Complete epsilon lemma statement

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}