Unbounded expansion of polynomials and products

Abstract

AbstractGiven $$d,s \in {\mathbb {N}}$$ d , s ∈ N , a finite set $$A \subseteq {\mathbb {Z}}$$ A ⊆ Z and polynomials $$\varphi _1, \dots , \varphi _{s} \in {\mathbb {Z}}[x]$$ φ 1 , ⋯ , φ s ∈ Z [ x ] such that $$1 \le \deg \varphi _i \le d$$ 1 ≤ deg φ i ≤ d for every $$1 \le i \le s$$ 1 ≤ i ≤ s , we prove that $$\begin{aligned} |A^{(s)}| + |\varphi _1(A) + \cdots + \varphi _s(A) | \gg _{s,d} |A|^{\eta _s}, \end{aligned}$$ | A ( s ) | + | φ 1 ( A ) + ⋯ + φ s ( A ) | ≫ s , d | A | η s , for some $$\eta _s \gg _{d} \log s / \log \log s$$ η s ≫ d log s / log log s . Moreover if $$\varphi _i(0) \ne 0$$ φ i ( 0 ) ≠ 0 for every $$1 \le i \le s$$ 1 ≤ i ≤ s , then $$\begin{aligned} |A^{(s)}| + |\varphi _1(A) \dots \varphi _s(A) | \gg _{s,d} |A|^{\eta _s}. \end{aligned}$$ | A ( s ) | + | φ 1 ( A ) ⋯ φ s ( A ) | ≫ 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.