The Tube Doubling Conjecture

Let $n\geq 2$ and $\varepsilon>0$. For sufficiently small $\delta>0$, let $\mathbb{T}$ be a set of $\delta$-tubes in $\mathbb{R}^n$, and let $\widetilde T$ denote the 2-fold dilate of $T$. The Tube Doubling Conjecture.

$$\left|\bigcup_{T\in\mathbb{T}}\widetilde T\right|\leq\delta^{-\varepsilon}\left|\bigcup_{T\in\mathbb{T}}T\right|.$$

This expresses that doubling the tubes increases the union volume by at most a sub-polynomial factor. It is known in dimension two, was open in dimensions three and higher in the surrounding discussion, and the paper proves it in $\mathbb{R}^3$.