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Nonbases of Density Zero not Contained in Maximal Nonbases
Author(s) -
Erdős Paul,
Nathanson Melvyn B.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-15.3.403
Subject(s) - zero (linguistics) , basis (linear algebra) , mathematics , integer (computer science) , combinatorics , sequence (biology) , discrete mathematics , computer science , geometry , chemistry , philosophy , linguistics , biochemistry , programming language
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.