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s ‐Maximal Nonbases of Density Zero
Author(s) -
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/s2-15.1.29
Subject(s) - mathematics , zero (linguistics) , integer (computer science) , combinatorics , basis (linear algebra) , class (philosophy) , order (exchange) , set (abstract data type) , function (biology) , zero set , discrete mathematics , geometry , computer science , artificial intelligence , philosophy , linguistics , finance , evolutionary biology , economics , biology , programming language
The set A 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 . The set A is a nonbasis if it is not a basis. Let A be a nonbasis, and let W be a finite set of non‐negative integers not belonging to A . Then A is an s ‐maximal nonbasis if A ∪ W remains a nonbasis for | W | < s , but A ∪ W becomes a basis for | W | = s . For every s ⩾ 1 we construct a class of s ‐maximal nonbases of density zero. Moreover, if A ( x ) denotes the counting function of A , then A ( x ) has order of magnitude √ x .