Block partitions in higher dimensions

Abstract

Consider a set $X\subseteq {\mathbb{R}}^d$ which is 1-dense, namely, it intersects every unit ball. We show that we can get from any point to any other point in ${\mathbb{R}}^d$ in $n$ steps so that the intermediate points are in $X$, and the discrepancy of the step vectors is at most $2\sqrt{2}$, or formally, $$\underset{\begin{array}{c}n\in {\mathbb{Z}}^{+},t\in {\mathbb{R}}^d\\ X\text{ is 1-dense}\end{array}}{\sup }\underset{\begin{array}{c}{p}_1,\dots ,p_{n-1}\in X\\ p_0=\underline{0},p_n=t\end{array}}{\inf }\underset{0\le i<j<n}{\max }\Vert (p_{i+1}-p_i)-(p_{j+1}-p_j)\Vert \le 2\sqrt{2}.$$