### Abstract

We consider a partition of the subsets of the natural numbers ℕ into two classes, the lower class and the upper class, according to whether the representation function of such a subset A, counting the number of pairs of elements of A whose sum is equal to a given integer, is bounded or unbounded. We give sufficient criteria for two subsets of ℕ to be in the same class and for a subset to be in the lower class or in the upper class.

Original language | English (US) |
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Title of host publication | Combinatorial and Additive Number Theory - CANT 2011 and 2012 |

Editors | Melvyn B. Nathanson |

Publisher | Springer New York LLC |

Pages | 43-53 |

Number of pages | 11 |

ISBN (Electronic) | 9781493916009 |

DOIs | |

State | Published - Jan 1 2014 |

Event | School on Combinatorics, Automata and Number Theory, CANT 2012 - Marseille, France Duration: May 21 2012 → May 25 2012 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 101 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | School on Combinatorics, Automata and Number Theory, CANT 2012 |
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Country | France |

City | Marseille |

Period | 5/21/12 → 5/25/12 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Haddad, L., & Helou, C. (2014). Lower and upper classes of natural numbers. In M. B. Nathanson (Ed.),

*Combinatorial and Additive Number Theory - CANT 2011 and 2012*(pp. 43-53). (Springer Proceedings in Mathematics and Statistics; Vol. 101). Springer New York LLC. https://doi.org/10.1007/978-1-4939-1601-6_4