Further bounds in the polynomial Szemerédi theorem

We show that there exists $c>0$ such that any subset of $\{1,\dots ,N\}$ having size $\gg N\mathrm{/}\exp ((\log \log \log N{)}^c)$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. 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.