Lattice point simplices

Abstract

We consider simplices in R m with lattice point vertices, no other boundary lattice points and n interior lattice points, with an emphasis on the barycentric coordinates of the interior points. We completely classify such triangles under unimodular equivalence and enumerate. For example, in a lattice point triangle with exactly one interior point, that point must be the centroid. We discuss the literature for fundamental tetrahedra and prove that there are seven possible barycentric coordinates for a one-point tetrahedron. Following suggestions of P. Erdös, we prove that, for fixed m and n, there are only finitely many possible sets of barycentric coordinates for the interior points. We also discuss a generalization of Beatty's problem in combinatorial number theory which has arisen several times in recent years.