Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube

Abstract

Higher-dimensional Dedekind sums are defined as a generalization of a recent one-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of diagonal lattice points leads to new formulae in certain special cases, and also to new bounds for the classical Dedekind sums. We define a new correspondence between n-dimensional Dedekind sums and certain convex n-dimensional cones, and we conjecture that these cones have a largest spacial angle of π / 6 . Bounds on n-dimensional Dedekind sums are important in the enumeration of lattice points in polytopes, since they are the building blocks for the lattice point enumerator of a polytope. Here, upper bounds for n-dimensional Dedekind sums are expressed in terms of 1-dimensional moments, and various relations among the moments are derived using statistical methods.