Unbounded expansion of polynomials and products

Abstract

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 \[ |A^{(s)}| + |φ_1(A) + \dots + φ_s(A) | \gg_{s,d} |A|^{η_s} , \] for some ${\eta }_s{\gg }_d\log s/\log \log s$. Moreover if ${\phi }_i(0)\ne 0$ for every $1\le i\le s$, then \[ |A^{(s)}| + |φ_1(A) \dots φ_s(A) | \gg_{s,d} |A|^{η_s}. \] 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.

arXiv comment28 pages, Revised version, To appear in Mathematische Annalen