$B_h$-sets of real and complex numbers

Abstract Let $K = \mathbf {R}$ or $\mathbf {C}$ . An n -element subset A of K is a $B_h$ -set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A . Associated with the $B_h$ -set $A = \{a_1,\ldots , a_n\}$ are the $B_h$ -vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$ . This article proves that “almost all” n -element subsets of K are $B_h$ -sets in the sense that the set of all $B_h$ -vectors is a dense open subset of $K^n$ .