Sharp growth for iterated sumsets of convex-function images

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

Background

The paper reviews known growth bounds for expressions formed from images f(A) and f(A-A) under repeated addition and subtraction. For convex functions, existing results establish strong growth for shorter expressions, while the paper proves improved bounds for certain analytic functions satisfying mutual-convexity conditions.

The stated conjecture seeks the natural k+1 exponent for all k-convex functions and all finite real sets A. The paper’s theorem for analytic functions with linearly independent derivatives does not establish this general conjectured bound.

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”