Unbounded expansion of polynomials and products

Given $d,s\in \mathbb{N}$, a finite set $A\subseteq \mathbb{Z}$ and polynomials ${\phi }_1,\dots ,{\phi }_s\in \mathbb{Z}[x]$ such that $1\le deg{\phi }_i\le d$ for every $1\le i\le s$, we prove that $$\mathrm{\mid }{A}^{(s)}\mathrm{\mid }+\mathrm{\mid }{\phi }_1(A)+\cdots +{\phi }_s(A)\mathrm{\mid }{\gg }_{s,d}\mathrm{\mid }A{\mathrm{\mid }}^{{\eta }_s},$$ for some ${\eta }_s{\gg }_d\log s\mathrm{/}\log \log s$. Moreover if ${\phi }_i(0)\mathrm{\ne }0$ for every $1\le i\le s$, then $$\mathrm{\mid }{A}^{(s)}\mathrm{\mid }+\mathrm{\mid }{\phi }_1(A)\dots {\phi }_s(A)\mathrm{\mid }{\gg }_{s,d}\mathrm{\mid }A{\mathrm{\mid }}^{{\eta }_s}\mathrm{.}$$ These generalise and strengthen previous results of Bourgain–Chang, Pálvölgyi–Zhelezov and Hanson–Roche-Newton–Zhelezov. We derive these estimates by proving the corresponding low-energy decompositions. The latter furnish further applications to various problems of a sum-product flavour, including questions concerning large additive and multiplicative Sidon sets in arbitrary sets of integers.