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 $$|2^{ n-1}f(A)-(2^{ n-1}-1)f(A)|\gg_{ \mathcal{ F},I} |A|^{ \phi(n)},$$ where $\phi:\mathbb{ N} \to \mathbb{ R}$ satisfies the recursive formula $$\phi(1)=1, \quad \phi(n)=1+\frac{ 1}{ 1+\frac{ 1}{ \phi(n-1)}} \quad \text{ for } n\geq 2.$$ The above result allows us to prove the bound $$|2^nf(A-A)-(2^n-1)f(A-A)| \gg_{ f,n,I} |A|^{ 1+\phi(n)}$$ where $f$ is an analytic function for which any $n$ distinct non-trivial discrete derivatives of $f'$ are linearly independent. This condition is satisfied, for instance, by any polynomial function of degree $m \geq n+1$. 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.
Categories: math.CO math.NT