### 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 |
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Pages (from-to) | 154-159 |

Number of pages | 6 |

Journal | Journal of Number Theory |

Volume | 12 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 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