On additive doubling and energy

TL;DR

This paper demonstrates that sets with small subtractive doubling contain large subsets with significantly high additive energy, revealing structural properties of such sets in abelian groups.

Contribution

It establishes a new link between small subtractive doubling and the existence of large subsets with high additive energy, with explicit bounds.

Findings

Large subsets with high additive energy exist within sets of small subtractive doubling.

The additive energy of these subsets exceeds a bound inversely related to the doubling constant.

The results hold with a universal exponent, specifically 1/37.

Abstract

We show that if A is a set having small subtractive doubling in an abelian group, that is |A-A|< K|A|, then there is a polynomially large subset B of A-A so that the additive energy of B is large than (1/K)^{1 - \epsilon) where epsilon is a positive, universal exponent. (1/37 seems to suffice.)