Popular differences for right isosceles triangles

For a subset $A$ of $\{1,2,\dots ,N{\}}^2$ of size $\alpha N^2$ we show existence of $(m,n)\mathrm{\ne }(0,0)$ such that the set $A$ contains at least $({\alpha }^3-o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best from arXiv:2101.02811. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field ${\mathbb{F}}_q$ we comment on exponential smallness of subsets of $({\mathbb{F}}_q^n{)}^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing proofs regarding three-term arithmetic progressions.