The Coin Exchange Problem and the Structure of Cube Tilings

Let k_1,…,k_d be positive integers, and D be a subset of [k_1]x…x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x…x{x_{s-1}}x[k_s]x{x_{s+1}}x…x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,…, k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.