Minimal bases and maximal nonbases in additive number theory |
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Authors: | Melvyn B Nathanson |
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Institution: | Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901 USA |
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Abstract: | An asymptotic basis of order h is minimal if no proper subset of is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of is a basis (resp. asymptotic basis) of order h, then is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h. |
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