ON MINIMAL ASYMPTOTIC BASES OF ORDER THREE

Let A be a subset of N, and W be a nonempty subset of N. Denote by F*(W) the set of all finite, nonempty subsets of W. For integer g >= 2, let A(g) (W) be the set of all numbers of the form Sigma(af)(f is an element of F)g(f) where F is an element of F*(W) and 1 <= a(f) <= g - 1. For i = 0; 1; 2, let W-i = {n is an element of N vertical bar n equivalent to i (mod 3)}. We show that for any g >= 2, the set A = A(g) (W-0) boolean OR A(g) (W-1) boolean OR A(g) (W-2) is a minimal asymptotic basis of order three. Moreover, we construct an asymptotic basis of order three containing no subset which is a minimal asymptotic basis of order three.