### Abstract

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let M_{h}
^{A} denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that M_{h}
^{A}(x) = O(x^{ 1-1 h+ε{lunate}}) for every ε{lunate} > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.

Original language | English |
---|---|

Pages (from-to) | 154-159 |

Number of pages | 6 |

Journal | Journal of Number Theory |

Volume | 12 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1980 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*12*(2), 154-159. https://doi.org/10.1016/0022-314X(80)90048-7

**Minimal asymptotic bases for the natural numbers.** / Erdős, P.; Nathanson, Melvyn B.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 12, no. 2, pp. 154-159. https://doi.org/10.1016/0022-314X(80)90048-7

}

TY - JOUR

T1 - Minimal asymptotic bases for the natural numbers

AU - Erdős, P.

AU - Nathanson, Melvyn B.

PY - 1980

Y1 - 1980

N2 - The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let Mh A denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that Mh A(x) = O(x 1-1 h+ε{lunate}) for every ε{lunate} > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.

AB - The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let Mh A denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that Mh A(x) = O(x 1-1 h+ε{lunate}) for every ε{lunate} > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.

UR - http://www.scopus.com/inward/record.url?scp=4243462999&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4243462999&partnerID=8YFLogxK

U2 - 10.1016/0022-314X(80)90048-7

DO - 10.1016/0022-314X(80)90048-7

M3 - Article

AN - SCOPUS:4243462999

VL - 12

SP - 154

EP - 159

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -