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Journal of the London Mathematical Society 1977 s2-15(3):403-405; doi:10.1112/jlms/s2-15.3.403
© 1977 by London Mathematical Society
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© Oxford University Press

Nonbases of Density Zero not Contained in Maximal Nonbases

Paul Erdos and Melvyn B. Nathanson

Mathematical Institute, Hungarian Academy of Sciences Budapest, Hungary
Department of Mathematics, Southern Illinois University Carbondale, Illinois 62901, U.S.A.

A sequence A = {ai} of non-negative integers is a basis if every sufficiently large integer n can be written in the form n = ai+aj with ai, aj isin A. If A is not a basis, then A is called a nonbasis. The nonbasis A is maximal if A {cup} {b} is a basis for every b {euro} A. We construct a nonbasis A of density zero, in particular, with A(x) = O({surd}x), such that A cannot be imbedded as a subset of any maximal nonbasis.


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