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The number of lattice points in irrational polytopes

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

doi:10.7282/t36d5w4j
fatcat:3pahuc7cuzacjfgkd7izuthgzy