Achievable Sets in Z^n

TL;DR

This paper investigates which subsets of Z^n can be expressed as differences of compact sets in R^n intersected with Z^n, using algebraic topology to analyze achievable sets and their properties.

Contribution

It introduces a graph-based method to characterize achievable sets in Z^n and extends results in two dimensions, advancing understanding of these sets for n ≥ 2.

Findings

Achievable sets correspond to connected components of associated graphs.

In two dimensions, the characterization of achievable sets is strengthened.

Open questions and generalizations are discussed using algebraic topology.

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.