New bound for Roth's theorem with generalized coefficients

Abstract

We prove the following conjecture of Shkredov and Solymosi: every subset $A \subset \mathbf{Z}^2$ such that $\sum_{a\in A\setminus\{0\}} 1/\left\|a\right\|^{2} = +\infty$ contains the three vertices of an isosceles right triangle. To do this, we adapt the proof of the recent breakthrough by Bloom and Sisask on sets without three-term arithmetic progressions, to handle more general equations of the form $T_1a_1+T_2a_2+T_3a_3 = 0$ in a finite abelian group $G$, where the $T_i$'s are automorphisms of $G$.