Corner-free sets via the torus

A corner is a triple of points in ${\mathbb{Z}}^2$ of the form $(x,y),(x+d,y),(x,y+d)$ where $d\mathrm{\ne }0$. One can think of them as being 2D-analogues to 3-term arithmetic progressions. In this short note, we extend ideas of Green-Wolf from this latter setting to the former, achieving slightly better constructions of corner-free sets.