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Essentialities in additive bases

Author: Peter Hegarty
Journal: Proc. Amer. Math. Soc. 137 (2009), 1657-1661
MSC (2000): Primary 11B13; Secondary 11B34
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.

References [Enhancements On Off] (What's this?)

  • [1] J. Cassaigne and A. Plagne, Grekos' $ S$ function has a linear growth, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2833-2840 (electronic). MR 2063100 (2005b:11011)
  • [2] B. Deschamps and B. Farhi, Essentialité dans les bases additives (French), J. Number Theory 123 (2007), 170-192. MR 2295438 (2008g:11016)
  • [3] P. Erdős, M. B. Nathanson and P. Tetali, Independence of solution sets and minimal asymptotic bases, Acta Arith. 69, no. 3 (1995), 243-258. MR 1316478 (96e:11014)
  • [4] M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory, J. Number Theory 6 (1974), 324-333. MR 0347764 (50:265)

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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

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.

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