A Danzer set for Axis Parallel Boxes

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 $O(T^d)$.

💡 Research Summary

The paper addresses a relaxed version of the classical Danzer problem, which asks for a discrete set D⊂ℝⁿ that meets every convex set of a fixed positive volume. Instead of all convex bodies, the authors consider only axis‑parallel boxes (aligned boxes) and define an “align‑Danzer set” as a discrete set that intersects every aligned box of a given volume s>0. Their main contribution is to exhibit explicit constructions of such sets in every dimension d≥2 that achieve the optimal growth rate O(Tᵈ), i.e., the number of points inside a ball of radius T grows proportionally to the volume of the ball.

The paper presents two constructions:

1. Two‑dimensional explicit construction (Theorem 1.1).
   The authors define a set D⊂ℝ² by taking all bi‑infinite binary sequences (aₙ)ₙ∈ℤ that contain only finitely many 1’s (denoted {0,1}^ℤ_Fin) and mapping each sequence to the point
   \