On $B_{h}[1]$-sets which are asymptotic bases of order $2h$

Abstract

Let h, k ≥ 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 [g]-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 [1] sets which are asymptotic bases of order 2h by using probabilistic methods.