Growth in Sumsets of Higher Convex Functions

Abstract

AbstractThe main results of this paper concern growth in sums of a k-convex function f. Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71–85, 2020)) of a growth result for f(A) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for $$\begin{aligned} |2^k f(A) - (2^k-1)f(A)|. \end{aligned}$$ | 2 k f ( A ) - ( 2 k - 1 ) f ( A ) | . We also generalise a recent result from Hanson et al. (J Lond Math Soc, 2021), proving that for any finite $$A\subset \mathbb {R}$$ A ⊂ R $$\begin{aligned} | 2^k f(sA-sA) - (2^k-1) f(sA-sA)| \gg _s |A|^{2s} \end{aligned}$$ | 2 k f ( s A - s A ) - ( 2 k - 1 ) f ( s A - s A ) | ≫ s | A | 2 s where $$s = \frac{k+1}{2}$$ s = k + 1 2 . This allows us to prove that, given any natural number $$s \in \mathbb {N}$$ s ∈ N , there exists $$m = m(s)$$ m = m ( s ) such that if $$A \subset \mathbb {R}$$ A ⊂ R , then $$\begin{aligned} |(sA-sA)^{(m)}| \gg _s |A|^{s}. \end{aligned}$$ | ( s A - s A ) ( m ) | ≫ s | A | s . This is progress towards a conjecture (Balog et al. in Electron J Comb 24(3):Paper No. 3.14, 17, 2017) which states that (1) can be replaced with $$\begin{aligned} |(A-A)^{(m)}| \gg _s |A|^{s}. \end{aligned}$$ | ( A - A ) ( m ) | ≫ s | A | s . Developing methods of Solymosi, and Bloom and Jones, and using an idea from Bradshaw et al. (Electron J Comb 29, 2021), we present some new sum-product type results in the complex numbers $$\mathbb {C}$$ C and in the function field $$\mathbb {F}_q((t^{-1}))$$ F q ( ( t - 1 ) ) .