Further bounds in the polynomial Szemerédi theorem

Abstract: We show that there exists $c &gt; 0$ such that any subset of ${1,\ldots,N}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y<sup>3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.