Achievable sets in Z(n)

作者：Sankar Krishanu Roy^*

来源：Forum Mathematicum, 2015, 27(1): 57-76.

DOI：10.1515/forum-2012-0004

• Geometric group theory
• number theory

What sets A subset of Z(n) can be written in the form (K - K) boolean AND 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 chara cterizes achievable sets for n = 1, but not much is known for n >= 2. We attempt to characterize achievable sets further by showing that with any finite, symmetric set A subset of Z(n) containing zero, we may associate a graph G(A). Then if A is achievable, we show that the set associated to some connected component of G(A) is achievable. In two dimensions, we can strengthen this theorem. Further generalizations and open questions are discussed. Throughout, the language and formalism of algebraic topology are useful.

• 出版日期2015-1
• 单位MIT