On Bh[1]-sets which are asymptotic bases of order 2h

Abstract

Let $h,k\ge 2$ be integers. A set A of positive integers is called asymptotic basis of order k if every large enough positive integer can be written as the sum of k terms from A. A set of positive integers A is said to be a $B_h\left[g\right]$-set if every positive integer can be written as the sum of h terms from A at most g different ways. In this paper we prove the existence of $B_h\left[1\right]$ sets which are asymptotic bases of order 2h by using probabilistic methods.