Additive growth amongst images of linearly independent analytic functions

Abstract

Let $\mathcal{F}$ be a set of $n$ real analytic functions with linearly independent derivatives restricted to a compact interval $I$. We show that for any finite set $A\subset I$, there is a function $f\in \mathcal{F}$ that satisfies $$|2^{n-1}f(A)-(2^{n-1}-1)f(A)|{\gg }_{\mathcal{F},I}|A{|}^{\varphi (n)},$$ where $\varphi :\mathbb{N}\to \mathbb{R}$ satisfies the recursive formula $$\varphi (1)=1,\varphi (n)=1+\frac{1}{1+\frac{1}{\varphi (n-1)}}\text{for }n\ge 2.$$ The above result allows us to prove the bound $$|2^{n}f(A-A)-(2^n-1)f(A-A)|{\gg }_{f,n,I}|A{|}^{1+\varphi (n)}$$ where $f$ is an analytic function for which any $n$ distinct non-trivial discrete derivatives of $f^{\prime }$ are linearly independent. This condition is satisfied, for instance, by any polynomial function of degree $m\ge n+1$. We also check this condition for the function $\arctan (e^x)$ with $n=3$, allowing us to improve upon a recent bound on the additive growth of the set of angles in a Cartesian product due to Roche-Newton.

22 pages. Resolved an inconsistency in terminology and fixed a small hole in the proofs of Theorems 1.8 and 1.9, amongst other clarifications and minor corrections