New bound for Roth's theorem with generalized coefficients

TL;DR

This paper proves that any subset of  with a divergent sum of inverse squared norms contains an isosceles right triangle, extending techniques from progressions to more general equations in finite abelian groups.

Contribution

It establishes a new bound for Roth's theorem with generalized coefficients by adapting recent progress on arithmetic progressions to broader equations.

Findings

Sets with divergent inverse squared norms contain the three vertices of an isosceles right triangle.

The proof adapts Bloom and Sisask's method to handle equations involving automorphisms in finite abelian groups.

The result confirms a conjecture of Shkredov and Solymosi for subsets of .

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 $G$.