Danzer set

One way to define the problem more formally is to consider the growth rate of a set ${\displaystyle S}$ in ${\displaystyle d}$-dimensional Euclidean space, defined as the function that maps a real number ${\displaystyle r}$ to the number of points of ${\displaystyle S}$ that are within distance ${\displaystyle r}$ of the origin. Danzer's question is whether it is possible for a Danzer set to have growth rate ${\displaystyle O(r^{d})}$, expressed in big O notation. If so, this would equal the growth rate of well-spaced point sets like the integer lattice (which is not a Danzer set).[1]

An equivalent formulation involves the density of a set ${\displaystyle S}$, defined as

$${\displaystyle \limsup _{r\to \infty }{\frac {|S\cap B_{d}(r)|}{V_{d}(r)}},}$$

where ${\displaystyle B_{d}(r)}$ denotes the Euclidean ball of radius ${\displaystyle r}$ in ${\displaystyle d}$-dimensional Euclidean space, centered at the origin, and ${\displaystyle V_{d}(r)}$ denotes its volume. Danzer's question asks whether there exists a Danzer set of bounded density or, alternatively, whether every set of bounded density has arbitrarily high-volume convex sets disjoint from it.[3]

Instead of asking for a set of bounded density that intersects arbitrary convex sets of unit volume, it is equivalent to ask for a set of bounded density that intersects all ellipsoids of unit volume, or all hyperrectangles of unit volume. For instance, in the plane, the shapes of these intersecting sets can be restricted to ellipses, or to rectangles. However, these shapes do not necessarily have their sides or axes parallel to the coordinate axes.[3]

It is possible to construct a Danzer set of growth rate that is within a polylogarithmic factor of ${\displaystyle O(r^{d})}$. For instance, overlaying rectangular grids whose cells have constant volume but differing aspect ratios can achieve a growth rate of ${\displaystyle O(n^{d}\log ^{d-1}n)}$.[4] Constructions for Danzer sets are known with a somewhat slower growth rate, ${\displaystyle O(r^{d}\log r)}$. However, because these growth rates are not ${\displaystyle O(r^{d})}$, these sets do not have bounded density, and the answer to Danzer's question remains unknown.[5][3]

Although the existence of a Danzer set of bounded density remains open, it is possible to restrict the classes of point sets that may be Danzer sets in other ways than by their densities, ruling out certain types of solution to Danzer's question. In particular, a Danzer set cannot be the union of finitely many lattices,[4] it cannot be generated by choosing a point in each tile of a substitution tiling (in the same position for each tile of the same type), and it cannot be generated by the cut-and-project method for constructing aperiodic tilings. Therefore, the vertices of the pinwheel tiling and Penrose tiling are not Danzer sets.[5]

• Heilbronn triangle problem, on sets of points that do not form triangles of small area
• Minkowski's theorem, that every unit-volume closed convex body that is centrally symmetric around the origin contains a nonzero point of the half-integer lattice