On the maximum density of minimal asymptotic bases

Abstract

A set A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer is the sum of h elements of A . It is proved that if A is an asymptotic basis of order h with lower asymptotic density dL (A) > /h , then there is a set W contained in A such that W has positive asymptotic density and A\W is an asymptotic basis of order h . This implies that if A is a minimal asymptotic basis of order h , then dL (A) 2. Minimal asymptotic bases form an important extremal class in additive number theory. The asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It follows that if A is minimal, then for every element a E A there must be infinitely many positive integers n, each of whose representations as a sum of h elements of A includes the number a as a summand. St6hr [7] introduced the concept of minimal asymptotic basis, and Hartter [3] proved that minimal asymptotic bases of order h exist for all h > 2. Erdos and Nathanson [1] survey recent results on minimal asymptotic bases. For any set A of integers, the counting function of A, denoted A(x), is defined by A(x) = card({a E Al 2, Nathanson [5, 6] has constructed minimal asymptotic bases A of order h that satisfy A(x) x for every Received by the editors December 12, 1986 and, in revised form, January 11, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 11 B 1 3, 1 1 B05, 11 P99.