The Coin Exchange Problem and the Structure of Cube Tilings
Abstract
Let k1,...,kd be positive integers, and D be a subset of [k1]x...x[kd], whose complement can be decomposed into disjoint sets of the form x1x...xxs-1x[ks]xxs+1x...xxd. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k1,..., kd with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.
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