Optimal long-sum growth for images of difference sets under k-convex functions

Prove that every finite subset A of the real numbers and every k-convex function f satisfy |2^k f(A-A)-(2^k-1)f(A-A)|\gg |A|^{k+1}.

Background

The paper explains that known results establish strong additive growth for images of sets under convex or higher-convex functions, but the corresponding optimal estimate for iterated sumsets of f(A-A) is not known in general. The conjectured exponent k+1 would extend the established three-term phenomenon to arbitrary k and would parallel the analogous theorem for higher-convex sets. The statement is attributed to Bradshaw and is explicitly presented as a conjecture.

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”