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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Essentialities in additive bases
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by Peter Hegarty PDF
Proc. Amer. Math. Soc. 137 (2009), 1657-1661 Request permission

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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2470824