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- W50219627 abstract "Let ]N denote the nonnegative integers, and let B~]N . Let hB denote the set of all sums of h not necessarily distinct elements of B . If hB = IN , then B is a basis of order h . If hB contains all but finitely many elements of I'4, then B is an asymptotic basis of order h . The set B is a minimal basis of order h if hB = IN, but hB' / IN for every proper subset B' ~ B. Similarly, B is a minimal asymptotic basis of order h if B is an asymptotic basis of order h , but no proper subset of B is an asymptotic basis of order h ; that is, hB contains all sufficiently large numbers, but for every b e B there are infinitely many integers n such that n ~ h(B~b}). StBhr [IZ] introduced this idea of minimality, and proved that every basis of order h contains a minimal basis of order h . H~[rtter [8] showed by a nonconstructive argument that there exist minimal asymptotic bases, and Nathanson [i0] constructed examples of minimal asymptotic bases of order h for all h_> Z. H~rtter [8] and Nathanson [I0] also gave examples of asymptotic bases of order Z that do not contain any minimal asymptotic basis of order Z. Indeed, ErdBs and Nathanson [3] showed that there exist asymptotic bases B of order Z such that, for every subset S~ B , if ISI < 00, then BS remains an asymptotic basis of order Z, but if ISI = o0, then BS is no longer an asymptotic basis of order Z. Clearly, B is an asymptotic basis that does not contain a minimal asymptotic basis. There is no classification of minimal asymptotic bases, nor is there any simple criterion to insure that an asymptotic basis contains a minirr~l asymptotic basis. Erd•s and Nathanson [6] proved that the square-free numbers contain a minimal asymptotic basis of order Z and also that there is an asymptotic basis of order Z of square-free numbers no subset of which is minimal. But it is usually difficult to determine whether or not a given asymptotic basis contains a minimal asymptotic basis. For example, the set B = {mZ+n Z}~,n=0 is a basis of order Z" @default.
- W50219627 created "2016-06-24" @default.
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- W50219627 date "1977-01-01" @default.
- W50219627 modified "2023-09-24" @default.
- W50219627 title "Oscillations of bases in number theory and combinatorics" @default.
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- W50219627 doi "https://doi.org/10.1007/bfb0063066" @default.
- W50219627 hasPublicationYear "1977" @default.
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