Given $d,s \in \mathbb{ N}$, a finite set $A \subseteq \mathbb{ Z}$ and polynomials $\varphi_1, \dots, \varphi_{ s} \in \mathbb{ Z}[x]$ such that $1 \leq deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{ (s)}| + |\varphi_1(A) + \dots + \varphi_s(A) | \gg_{ s,d} |A|^{ \eta_s} , \] for some $\eta_s \gg_{ d} \log s / \log \log s$. Moreover if $\varphi_i(0) \neq 0$ for every $1 \leq i \leq s$, then \[ |A^{ (s)}| + |\varphi_1(A) \dots \varphi_s(A) | \gg_{ s,d} |A|^{ \eta_s}. \] These generalise and strengthen previous results of Bourgain--Chang, P\'{ a}lv\"{ o}lgyi--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.
Categories: math.NT math.CO