Tube Doubling Conjecture in higher dimensions (n ≥ 4)

Prove that for every ε > 0 and all sufficiently small δ > 0, any set 𝒯 of δ-tubes in R^n (for dimensions n ≥ 4) satisfies the doubling inequality |⋃_{T∈𝒯} \tilde T| ≤ δ^{-ε} |⋃_{T∈𝒯} T|, where \tilde T denotes the 2-fold dilate of T.

Background

The paper states the Tube Doubling Conjecture and notes it was known in dimension two and open in three and higher dimensions. The authors then prove it in R^3, leaving higher dimensions unresolved.

This conjecture is closely connected to multiplier problems and geometric combinatorics, and generalizing the three-dimensional resolution to higher dimensions remains a significant challenge.