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Essentialities in additive bases.
- Source :
-
Proceedings of the American Mathematical Society . Dec2008, Vol. 137 Issue 5, p1657-1661. 5p. - Publication Year :
- 2008
-
Abstract
- Let $A$ be an asymptotic basis for $mathbb {N}_0$ of some order. By an {em essentiality} of $A$ one means a subset $P$ such that $A backslash P$ is no longer an asymptotic basis of any order and such that $P$ is minimal among all subsets of $A$ with this property. A finite essentiality of $A$ is called an {em essential subset}. In a recent paper, Deschamps and Farhi asked the following two questions: (i) Does every asymptotic basis of $mathbb {N}_0$ possess some essentiality? (ii) Is the number of essential subsets of size at most $k$ of an asymptotic basis of order $h$ (a number they showed to be always finite) bounded by a function of $k$ and $h$ only? We answer the latter question in the affirmative and answer the former in the negative by means of an explicit construction, for every integer $h geq 2$, of an asymptotic basis of order $h$ with no essentialities. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 137
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 36177260
- Full Text :
- https://doi.org/10.1090/S0002-9939-08-09732-3