A Danzer set for axis parallel boxes (Q2796746)

The starting point of the authors is the question whether a discrete Danzer set in \(\mathbb{R}^d\) of growth rate \(O(T^n)\) exists. Remember that \(D\subseteq \mathbb{R}^n\) is a Danzer set whenever, for some \(s>0\), \(D\) intersects every convex set of volume \(s\). Their approach to this problem consists of considering a smaller family of sets, the align-Danzer sets (intersecting every aligned box \([a_1,b_1]\times\cdots\times[a_n,b_n]\) of volume \(s\)). In this setting, they provide concrete constructions of discrete align-Denzer sets having growth rate \(O(T^n)\).NEWLINENEWLINEWhen \(n=2\) they give (Theorem 1.1) an align-Danzer set which is a variant of the binary version of van der Corput sequence (see [\textit{J. G. van der Corput}, Proc. Akad. Wet. Amsterdam 38, 813--821 (1935; Zbl 0012.34705)]), having growth rate \(O(T^2).\) According to the authors, ``although the set \(D\) in Theorem 1.1 is given very explicitly, and the proof is by elementary means, it only solves the problem in dimension \(2\), and no simple higher-dimensional extension comes to mind'', so in higher dimensions (Theorem 1.3) they opt for an approach based on admissible lattices in \(\mathbb R^n.\) Again, the construction is not new (in fact, their Theorem 1.3 is immediate from Theorem 1.2 in [\textit{M. M. Skriganov}, Invent. Math. 132, No. 1, 1--72 (1998; Zbl 0902.11041)]), but their merit relies on the fact of connecting these sets with Danzer's problem. Finally, applying the above results they reprove a result in computational geometry, namely, Corollary 1.4: For every \(\varepsilon >0\) there are \(\varepsilon\)-nets of optimal sizes \(O(1/\varepsilon)\) for the range space \((X,\mathcal{R})\), being \(X\) the \(n\)-dimensional cube and \(\mathcal{R}\) the set of aligned boxes.