Danzer's Problem

Question: does there exist a set S ⊂ ℝ^d hitting every convex body of volume 1 (|S ∩ K| ≥ c > 0 for every such K), whose points grow at most linearly — |S ∩ \(B_{R}\)| = O(\(R^{d}\))? Such an S is called a Danzer set.

Part of the same 1960s wave as the packing and covering classics on our shelves. Lattice-like sets hit every large body but fail volume-1 bodies; probabilistic constructions hit everything with exponentially growing counting functions. Solomon and Weiss (2018) built Danzer sets under a dynamical hypothesis, and Král and collaborators pushed related quantitative versions — but the linear-growth question itself stands untouched. It quietly controls deep questions about quasicrystals and Delone-set Fourier analysis.