Popular Differences for Corners in Abelian Groups

Abstract

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 Szemerédi 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 $A\subset G\times G$ of density $\alpha$, one can fix $d\ne 0$ such that $A$ contains a large fraction, now known to be approximately ${\alpha }^4$, 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 ${\mathbb{Z}}^2$.

arXiv comment20 pages + references

Cited by in corpus (1)

• Khintchine-type recurrence for 3-point configurations