No abstract

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We study the number of lattice points in integer dilates of the rational polytope P ¼ ðx 1 ; . . . ; x n Þ 2 R n 50 :where a 1 ; . . . ; a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1 ; . . . ; a n ; find the largest value of t (the Frobenius number) such that m 1 a 1 þ Á Á Á þ m n a n ¼ t has no solution in positive integers m 1 ; . . . ; m n : This is equivalent to the problem of finding the largest dilate tP such that the facet f P n k¼1 x k a k ¼ tg contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials Lð % P P; tÞ :¼ #ðtP \ Z n Þ and LðP8; tÞ :¼ #ðtP8 \ Z n Þ: Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier-Dedekind sums and use them to give new bounds for the Frobenius number. # 2002 Elsevier Science (USA)

We study the problem of covering R d by overlapping translates of a convex body P , such that almost every point of R d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov [3] and Minkowski [11]. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile R d for some positive integer k.