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**Harvard**

Hegarty, P. (2009) *Essentialities in additive bases*.

** BibTeX **

@article{

Hegarty2009,

author={Hegarty, Peter},

title={Essentialities in additive bases},

journal={Proceedings of the American Mathematical Society},

issn={0002-9939},

volume={137},

issue={5},

pages={1657-1661},

abstract={Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\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 N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.},

year={2009},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 75261

A1 Hegarty, Peter

T1 Essentialities in additive bases

YR 2009

JF Proceedings of the American Mathematical Society

SN 0002-9939

VO 137

IS 5

SP 1657

OP 1661

AB Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\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 N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities.

LA eng

DO 10.1090/S0002-9939-08-09732-3

LK http://dx.doi.org/10.1090/S0002-9939-08-09732-3

OL 30