Popular Differences for Corners in Abelian Groups

For a compact abelian group $G$, a corner in $G\times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemer'edi implies that for every $\alpha >0$, there is some $\delta >0$ such that every subset $A\subset G\times G$ of density $\alpha$ contains a $\delta$ fraction of all corners in $G\times G$, as $x,y,d$ range over $G$. Recently, Mandache proved a “popular differences” version of this result in the finite field case $G={\mathbb{F}}_p^n$, showing that for any subset $A \subset G \times G$ofdensity$\alpha$,onecanfix$d \neq 0$suchthat$A$containsalargefraction,nowknowntobeapproximately$\alpha^4$, of all corners with difference $d$, as $x,y$ vary over $G$. We generalize Mandache’s result to all compact abelian groups $G$, as well as the case of corners in ${\mathbb{Z}}^2$.