Effective bounds for Roth's theorem with shifted square common difference

Abstract

Let $S$ be a subset of $\{1,\dots ,N\}$ avoiding the nontrivial progressions $x,x+y^2-1,x+2(y^2-1)$. We prove that $|S|\ll N/{\log }_{m}N$, where ${\log }_m$ is the $m$-fold iterated logarithm and $m\in \mathbf{N}$ is an absolute constant. This answers a question of Green.

47 pages