Optimal sumset and difference-set growth from control

Prove that every finite subset A of an abelian group with control κ satisfies |A−A| ≫_ε κ^{−1+ε}|A| and |A+A| ≫_ε κ^{−1+ε}|A| for every ε>0.

Background

The paper introduces control κ as an L^3 convolutional measure of additive structure. Existing results give lower bounds for both the sumset and difference set that improve on trivial estimates but do not reach the conjectured exponent of κ^{-1}.

The conjecture would imply near-optimal additive growth for important examples such as convex sets, whose control is on the order of |A|^{-1}. The paper derives conditional consequences of this conjecture for convex sets and convex-function configurations.