ALIBABA 2021 Problem 10

In $\mathbb{R}^3$ , for a rectangular box $\Delta$ , let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$ . For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$ ), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$ .

\medskip

If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$ , we say that they are almost identical.

Find the largest real number $a$ such that the following holds for some $C=C(a)>0$ :

For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$ , assuming that (i) $\vert S\vert=N$ , (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$ -plane. Then the volume $$\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.$$