The number of lattice points in irrational polytopes

Bence Borda
2016
The discrepancy | t P ∩ Z^d | - lambda (P) t^d is studied as a function of the real variable t>1, where P is a polytope in R^d with at least one vertex not in Z^d. For a special class of cross polytopes and orthogonal simplices defined in terms of algebraic numbers the discrepancy is proved to be the sum of an explicit polynomial of t and a randomly fluctuating term of smaller order of magnitude. This phenomenon has not yet been described in the literature for any convex body. A general
more » ... . A general discrepancy bound is proved for polytopes the coordinates of the vertices of which all belong to a given real quadratic field. As a corollary a new property of the regular dodecahedron is obtained. Finally, answering a question of Beck, a formula is given for the variance of a random fluctuation arising in a lattice point counting problem on the plane. The main methods used are Fourier analysis and the theory of Diophantine approximation.
doi:10.7282/t36d5w4j fatcat:3pahuc7cuzacjfgkd7izuthgzy