Correct energy exponent for sets with control

Determine the optimal exponent c such that every finite set A with control parameter κ satisfies an upper bound of the form E(A)≪ε κ^c|A|³, where E(A) denotes additive energy, and in particular determine whether the exponent 0.7549… is sharp.

Background

The paper explains that the naive conjecture E(A)≪ε κ^1−ε|A|³ cannot hold in general because of examples connected with the Balog–Szemerédi–Gowers theorem. Those examples show that the exponent cannot exceed approximately 0.7549.

The authors do not establish whether this obstruction gives the true optimal exponent. Resolving the question would clarify the strongest possible relationship between L³ control and additive energy and would improve the quantitative Balog–Szemerédi–Gowers theorem if the optimal exponent were smaller.