Generalize the trilinear Fourier restriction approach to cinematic Furstenberg sets

Determine whether the trilinear Fourier restriction approach used for circular Furstenberg sets can be generalized to cinematic (u, v)-Furstenberg sets, including whether such a generalization is possible using variable-coefficient trilinear Fourier restriction.

Background

The paper proves incidence bounds and dimension estimates for circular and sine-wave Furstenberg sets, and also obtains certain bounds for the broader class of cinematic Furstenberg sets. The proof of one circular bound uses a trilinear Fourier restriction theorem specialized to the cone, whereas the cinematic-curve arguments developed later rely on variable-coefficient local smoothing and do not establish an analogous trilinear estimate.

The authors observe that variable-coefficient trilinear Fourier restriction results might provide a route to such an extension, but explicitly state that they have not carried it out and leave its feasibility unresolved. This is a concrete methodological open question concerning whether the stronger trilinear approach can yield corresponding results for cinematic families.