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 | References | Similar Articles | Additional Information

Abstract: Let be an asymptotic basis for of some order. By an *essentiality* of one means a subset such that is no longer an asymptotic basis of any order and such that is minimal among all subsets of with this property. A finite essentiality of is called an *essential subset*. In a recent paper, Deschamps and Farhi asked the following two questions: (i) Does every asymptotic basis of possess some essentiality? (ii) Is the number of essential subsets of size at most of an asymptotic basis of order (a number they showed to be always finite) bounded by a function of and 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 , of an asymptotic basis of order with no essentialities.

**[1]**Julien Cassaigne and Alain Plagne,*Grekos’ 𝑆 function has a linear growth*, Proc. Amer. Math. Soc.**132**(2004), no. 10, 2833–2840 (electronic). MR**2063100**, 10.1090/S0002-9939-04-07344-7**[2]**Bruno Deschamps and Bakir Farhi,*Essentialité dans les bases additives*, J. Number Theory**123**(2007), no. 1, 170–192 (French, with English and French summaries). MR**2295438**, 10.1016/j.jnt.2006.06.002**[3]**Paul Erdős, Melvyn B. Nathanson, and Prasad Tetali,*Independence of solution sets and minimal asymptotic bases*, Acta Arith.**69**(1995), no. 3, 243–258. MR**1316478****[4]**Melvyn B. Nathanson,*Minimal bases and maximal nonbases in additive number theory*, J. Number Theory**6**(1974), 324–333. MR**0347764**

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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

DOI:
https://doi.org/10.1090/S0002-9939-08-09732-3

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.