Correction Q->R plus Sebi wording suggestion

main.tex

@@ -10,6 +10,7 @@
 
 \newcommand{\QQ}{\mathbb{Q}}
 \newcommand{\ZZ}{\mathbb{Z}}
+\newcommand{\RR}{\mathbb{R}}
 \newcommand{\chern}{\operatorname{ch}}
 \newcommand{\lcm}{\operatorname{lcm}}
 \newcommand{\gcd}{\operatorname{gcd}}
@@ -28,12 +29,12 @@ Practical Methods for Finding Pseudowalls}
 
 \section{Introduction}
 
-[ref] shows that for any $\beta_0 \in \QQ$,
-the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \QQ_{>0}\}$ only
+[ref] shows that for any rational $\beta_0$,
+the vertical line $\{\sigma_{\alpha,\beta_0} \colon \alpha \in \RR_{>0}\}$ only
 intersects finitely many walls. A consequence of this is that if
-$\beta_{-} \in \QQ$, then there can only be finitely many circular walls to the
+$\beta_{-}$ is rational, then there can only be finitely many circular walls to the
 left of the vertical wall $\beta = \mu$.
-On the other hand, when $\beta_{-} \not\in \QQ$, [ref] showed that there are
+On the other hand, when $\beta_{-}$ is not rational, [ref] showed that there are
 infinitely many walls.
 
 This dichotomy does not only hold for real walls, realised by actual objects in
@@ -43,11 +44,11 @@ which satisfy certain numerical conditions which would be satisfied by any real
 destabilizer, regardless of whether they are realised by actual elements of
 $\bddderived(X)$.
 
-Since real walls are a subset of pseudowalls, the $\beta_{-} \not\in \QQ$ case
+Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
 follows immediately from the corresponding case for real walls.
-However, the $\beta_{-} \in \QQ$ case involves showing that the following
+However, the rational $\beta_{-}$ case involves showing that the following
 conditions only admit finitely many solutions (despite the fact that the same
-conditions admit infinitely many solutions when $\beta_{-} \not\in \QQ$).
+conditions admit infinitely many solutions when $\beta_{-}$ is irrational).
 
 
 For a destabilizing sequence