Multi-dimensional Versions of a Theorem of Fine and Wilf and a Formula of Sylvester

Let ~ v0, ..., ~ vk be vectors in Z k which generate Zk. We show that a body V ⊂ Zk with the vectors ~ v0, ..., ~ vk as edge vectors is an almost minimal set with the property that every function f : V → R with periods ~ v0, ..., ~ vk is constant. For k = 1 the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma. Suppose ~0 is not a non-trivial linear combination of ~ v0, ..., ~ vk with nonnegative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of ~ v0, ..., ~ vk over Z≥0, but infinitely many points on each of its hyperfaces are not. For k = 1 the result reduces to a formula of Sylvester corresponding to Frobenius’ Coin-changing Problem in the case of coins of two denominations.