Oscillations of bases for the natural numbers

Abstract

Let A A be a set of positive integers. Then A A is a basis if every sufficiently large integer n n can be written in the form n = a i + a j n = {a_i} + {a_j} with a i , a j ϵ A {a_i},\;{a_j}\epsilon A . Otherwise, A A is a nonbasis. In this paper we construct sets which oscillate from basis to nonbasis to basis or from nonbasis to basis to nonbasis under finite perturbations of the sets.