Generalization of the trilinear 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, potentially using variable-coefficient trilinear Fourier restriction, and establish the resulting incidence or dimension bounds if such a generalization is possible.

Background

The paper proves incidence bounds for circles and sine waves using a trilinear Fourier restriction theorem and applies these bounds to obtain lower bounds for the Hausdorff dimension of circular and sine-wave Furstenberg sets. Cinematic Furstenberg sets are a broader class associated with families of curves satisfying cinematic curvature conditions.

The authors note that variable-coefficient trilinear Fourier restriction results may provide a route to extending this method to cinematic curves, but they explicitly state that they have not carried out the generalization and leave its feasibility unresolved. This is distinct from the paper’s established cinematic results, which use variable-coefficient local smoothing rather than the trilinear restriction method.