The Frobenius problem on lattices

It is widely known that if p and q are relatively prime positive integers then (a) the set of linear combinations of p and q with nonnegative integer coefficients includes all integers greater than pq − p − q, (b) exactly half the integers between 0 and pq − p − q belong to this set and (c) an integer m belongs to this set if and only if pq−p−q−m does not. A multidimensional version of statement (a) was recently obtained by Simpson and Tijdeman subject to a geometric condition. Here we unconditionally obtain multidimensional versions of all three statements.