Achievable Sets in Z^n

Abstract

What sets A \subset Z^n can be written in the form (K-K) \cap Z^n, where K is a compact subset of R^n such that K+Z^n=R^n? Such sets A are called achievable, and it is known that if A is achievable, then < A >=Z^n. This condition completely characterizes achievable sets for n=1, but not much is known for n \ge 2. We attempt to characterize achievable sets further by showing that with any finite, symmetric set A \subset Z^n containing zero, we may associate a graph G(A). Then if A is achievable, we show the set associated to some connected component of G(A) is achievable. In two dimensions, we can strengthen this theorem further. Further generalizations and open questions are discussed. Throughout, the language and formalism of algebraic topology are useful.