Question: does there exist a set \(S \subset \mathbb{R}^d\) hitting every convex body of volume 1 (\(|S \cap K| \geq c > 0\) for every such \(K\)), whose points grow at most linearly — \(|S \cap B_R| = O(R^d)\)? 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.
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