Growth in Sumsets of Higher Convex Functions

Abstract

The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof 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 \[ |2^k f(A) - (2^k-1)f(A)|. \] We also generalise a recent result of Hanson, Roche-Newton and Senger, by proving that for any finite $A\subset \mathbb{R}$ \[ | 2^k f(sA-sA) - (2^k-1) f(sA-sA)| \gg_s |A|^{2s} \] where $s = \frac{k+1}{2}$. This allows us to prove that, given any natural number $s \in \mathbb{N}$, there exists $m = m(s)$ such that if $A \subset \mathbb{R}$, then \begin{equation}\label{conj A-Aus} |(sA-sA)^{(m)}| \gg_s |A|^{s}. \end{equation} This is progress towards a conjecture which states that the above inequality can be replaced with \[|(A-A)^{(m)}| \gg_s |A|^{s}.\] Developing methods of Solymosi, and Bloom and Jones, we present some new sum-product type results in the complex numbers $\mathbb{C}$ and in the function field $\mathbb{F}_q((t^{-1}))$.