Efficiency of shading in Kakeya maximal function approaches in R^3

Investigate whether the Wang–Zahl framework can be refined to avoid iterative losses from λ to λ^2 in local density estimates, thereby maintaining a near-linear dependence on λ through rescalings and induction on scale and advancing toward the Kakeya maximal function conjecture in R^3.

Background

Progress toward the Kakeya maximal function in R^3 requires controlling shaded tube unions Y(T) with |Y(T)| ≥ λ|T|. Cordoba-type 2D arguments typically yield λ^2 density in local prisms, and repeated rescalings cause λ to degrade to λ^{2^N}, obstructing iteration.

The open problem asks for more efficient implementations (or stronger shading hypotheses) in the Wang–Zahl strategy that prevent such losses, potentially enabling induction-on-scale arguments compatible with Kakeya maximal function bounds.