Kakeya L^p conjecture for tube overlaps

Prove that for dimension d ≥ 2, any family of translates x_θ of unit-radius, length-R tubes θ* satisfies ∫ |∑_θ 1_{θ* + x_θ}(x)|^p dx ≤ C(ε,d) R^ε (R^{(d−1)p} + R^d) for all p ≥ 1 and ε > 0.

Background

This L^p formulation quantifies the maximal overlap of tubes in all directions, matching the trivial arrangement (all tubes centered at the origin) up to R^ε. It is another central form of the Kakeya problem with strong implications in harmonic analysis.

Resolving it would constrain tube overlaps and therefore improve restriction and decoupling estimates, hence impacting large-value bounds for structured matrices.