Nonbases of Density Zero not Contained in Maximal Nonbases

A sequence A = { a i } of non‐negative integers is a basis if every sufficiently large integer n can be written in the form n = a i + a j with a i , a j ∈ A . If A is not a basis, then A is called a nonbasis. The nonbasis A is maximal if A ∪ { b } is a basis for every b € A . We construct a nonbasis A of density zero, in particular, with A ( x ) = O (√ x ), such that A cannot be imbedded as a subset of any maximal nonbasis.