Dense minimal asymptotic bases of order two

Abstract

We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every h ⩾ 2 there exists a minimal asymptotic basis A of order h with d ( A ) = 1 / h , where d ( A ) denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order h ⩾ 2 such that A ( k ) = k / h + O ( 1 ) . Moreover, they asked if there exists a minimal asymptotic basis with lim sup ( a i + 1 − a i ) = 3 . In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition 1 2 k ⩽ A ( k ) ⩽ 1 2 k + 1 .