Sum Complexes and Uncertainty Numbers

Abstract

Let p be a prime and let A be a subset of F_p. For k<p let X_{A,k} be the (k-1)-dimensional complex on the vertex set F_p with a full (k-2)-skeleton whose (k-1)-faces are k-subsets S of F_p such that the sum of the elements of S belongs to A. The homology groups of X_{A,k} with field coefficients are determined. In particular it is shown that if |A| \leq k then H_{k-1}(X_{A,k};F_p)=0. This implies a homological characterization of uncertainty numbers of subsets of F_p.