Optimal higher-order growth for iterated sumsets of f(A-A)

Prove that every k-convex function f and every finite set A\subset R satisfy |2^k f(A-A)-(2^k-1)f(A-A)|\gg |A|^{k+1} for all k\in\mathbb N.

Background

The paper presents a conjecture attributed to Bradshaw concerning the extension of known cubic and higher-order growth results from f(A) to f(A-A). The conjectured exponent k+1 would match the natural higher-convexity growth pattern and substantially improve the bounds available for general classes of functions.

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”