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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Essentialities in additive bases


Author: Peter Hegarty
Journal: Proc. Amer. Math. Soc. 137 (2009), 1657-1661
MSC (2000): Primary 11B13; Secondary 11B34
DOI: https://doi.org/10.1090/S0002-9939-08-09732-3
Published electronically: December 17, 2008
MathSciNet review: 2470824
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Abstract: Let $A$ be an asymptotic basis for $\mathbb {N}_0$ of some order. By an 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 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.


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Additional Information

Peter Hegarty
Affiliation: Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and University of Gothenburg, SE-41296 Gothenburg, Sweden
Email: hegarty@math.chalmers.se

Keywords: Additive basis, essential subset.
Received by editor(s): March 10, 2008
Received by editor(s) in revised form: August 19, 2008
Published electronically: December 17, 2008
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.