A Danzer set for Axis Parallel Boxes

TL;DR

This paper constructs discrete sets in multi-dimensional space that intersect all axis-aligned boxes of volume one, achieving optimal growth rate, which has implications for geometric covering and sampling problems.

Contribution

It provides explicit constructions of Danzer sets in higher dimensions with optimal growth rate, advancing the understanding of geometric set coverage.

Findings

Constructed explicit Danzer sets in $ ^d$ for $d extgreater 1$

Sets intersect all volume-one axis-aligned boxes

Achieved optimal growth rate of $O(T^d)$

Abstract

We present concrete constructions of discrete sets in $\mathbb{R}^d$ ($d\ge 2$) that intersect every aligned box of volume $1$ in $\mathbb{R}^d$, and which have optimal growth rate $O(T^d)$.