A Salem-Spencer-Type Construction for Large Subsets of Integer Grids with No Isosceles Right Triangles

Let $F(n)$ be the largest size of a subset of $\{0,1,\dots ,n-1{\}}^2$ containing no nondegenerate isosceles right triangle. We give a modified Salem–Spencer-type construction over the Gaussian integers showing that $F(n)=\mathrm{\Omega }(n^{1.3})$. The best known upper bound is $F(n)\ll n^2\mathrm{/}(\log n{)}^{1+c}$ for some absolute constant $c>0$, so there is still a large gap between the bounds.