Optimal iterated growth for images of difference sets

Prove that for every k-convex function f, every finite subset A of the real numbers, and every positive integer k, the iterated sum-difference set satisfies |2^k f(A-A)-(2^k-1)f(A-A)| \gg |A|^{k+1}.

Background

The paper explains that longer sum-difference expressions of images of convex sets exhibit enhanced additive growth. For f(A-A), existing results give weaker exponents for general classes of functions, while the paper proves stronger bounds for analytic functions whose discrete derivatives satisfy mutual convexity conditions. The cited conjecture predicts the natural exponent k+1 for every k-convex function and every finite real set, substantially strengthening the currently established general theory.

References

Indeed, Bradshaw conjectures that for any $k$-convex function $f$, $$|2kf(A-A)-(2k-1)f(A-A)| \gg |A|{k+1}$$ should hold for all finite subsets $A \subset R$ and $k \in \mathbb{N}$.

— Additive growth amongst images of linearly independent analytic functions  (2503.03690 - Mansfield, 5 Mar 2025) in Section 1, subsection “Growth for sum sets of convex functions,” paragraph preceding Theorem \ref{f(A-A})