Steinhaus' lattice point problem for polyhedra (Q2833658)

In a previous paper [Discrete Math. 338, No. 3, 164--167 (2015; Zbl 1305.52020)], the author proved that any polygon (not necessarily convex) with area \(n\) has a congruent copy that contains exactly \(n\) integer lattice points; here \textit{congruent} means that the copy is a translate of the original, followed by a rotation. This solves the polygonal version of a problem about lattice points in circles, posed by Steinhaus in the 1950's.NEWLINENEWLINEThe present paper generalizes the above result to higher dimensions. The main result says that any \(d\)-dimensional polyhedron (by which the author means a compact set bounded by a \((d-1)\)-dimensional closed manifold that is contained in the union of finitely many hyperplanes in \(\mathbb R^d\)) with volume \(n + \alpha\), for some \(|\alpha| < 1\), has a congruent copy that contains exactly \(n\) points in~\(\mathbb Z^d\).