Correct exponent for additive energy under control

Determine whether the exponent 0.7549… is the correct exponent c in an upper bound E(A) ≪_ε κ^{c−ε}|A|^3 for finite sets A with control κ.

Background

The paper explains that the naive conjecture E(A) ≪_ε κ^{1−ε}|A|^3 cannot hold in general because constructions related to the Balog–Szemerédi–Gowers theorem impose an obstruction. The largest exponent currently compatible with the discussed obstruction is approximately 0.7549.

The authors explicitly state that it is unresolved whether this obstruction gives the true exponent, and note that further examples of controlled sets with large additive energy would be valuable.