Growth in Sumsets of Higher Convex Functions

TL;DR

This paper investigates the growth properties of sumsets involving higher convex functions, improving bounds, generalizing recent results, and advancing towards conjectures on sumset growth in real numbers, complex numbers, and function fields.

Contribution

It provides improved bounds for sumsets of convex functions, generalizes recent results, and makes progress on conjectures about sumset growth in various mathematical settings.

Findings

Improved bounds for sumsets of convex functions without logarithmic factors.

Generalization of recent sumset growth results to higher convex functions.

Progress towards conjectures on sumset growth in real numbers, complex numbers, and function fields.

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 $\mathbb{F}_q((t^{-1}))$.