Additive growth amongst images of linearly independent analytic functions
Samuel Mansfield
TL;DR
This work develops a framework to prove additive growth for images and iterated sum-difference sets under real-analytic, linearly independent derivative families. By introducing 1- and $k$-independence of analytic functions and leveraging squeezing, equidistribution, and discrete-derivative analysis (with Hurwitz’s theorem to control zeros), the authors derive a recursive bound $|2^{n-1}f(A)-(2^{n-1}-1)f(A)| \gg |A|^{\phi(n)}$ with $\phi(1)=1$ and $\phi(n)=1+1/(1+1/\phi(n-1))$, approaching the golden ratio $\phi=(1+\sqrt{5})/2$. They extend these results to $|2^n f(A-A)-(2^n-1)f(A-A)|$ under a flat-discrete-derivative independence condition, showing growth for polynomials of degree $m\ge n+1$ and for $f=\arctan(e^x)$ with $n=3$, improving bounds on the additive growth of angles in Cartesian products. The approach unifies convex-function growth with a broader class of analytic functions and yields new growth bounds for both set-analytic and geometric (angle) configurations, with clear implications for additive combinatorics and geometric incidence problems.
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 $$|2^{n-1}f(A)-(2^{n-1}-1)f(A)|\gg_{\mathcal{F},I} |A|^{φ(n)},$$ where $φ:\mathbb{N} \to \mathbb{R}$ satisfies the recursive formula $$φ(1)=1, \quad φ(n)=1+\frac{1}{1+\frac{1}{φ(n-1)}} \quad \text{for } n\geq 2.$$ The above result allows us to prove the bound $$|2^nf(A-A)-(2^n-1)f(A-A)| \gg_{f,n,I} |A|^{1+φ(n)}$$ 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.