Cofinite subsets of asymptotic bases for the positive integers

Abstract

Let A be an infinite set of integers containing at most finitely many negative terms. Let hA denote the set of all integers n such that n is a sum of h elements of A. Let F be a finite subset of A. Theorem. If hA contains an infinite arithmetic progression with difference d, and if gcd{a−a′¦a,a′∈A\\F} = d , then there exists q such that q( A\\ F) contains an infinite arithmetic progression. In particular, if A is an asymptotic basis, then A\\ F is an asymptotic basis if and only if gcd{ a− a′| a, a′∈ A\\ F} = 1. Theorem. If A is an asymptotic basis of order h, and if F⊆ A, card( F) = k, and A\\ F is an asymptotic basis, then the exact order of A\\ F is O( h k+1 ).