### Abstract

Using Feferman-Vaught techniques we show a certain property of the line spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that ifit is violated then one can find an admissible class with the same fine spectrum which docs not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, i.e., when the sizes of the structures in an admissible class K are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher s Axiom A″. The third condition is also for the discrete case, when there is a uniform bound on the number of K-indecomposables of any given size.

Original language | English |
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Pages (from-to) | 468-498 |

Number of pages | 31 |

Journal | Canadian Journal of Mathematics |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 1997 |

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### Keywords

- First order limit laws
- Generalized number theory

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Canadian Journal of Mathematics*,

*49*(3), 468-498. https://doi.org/10.4153/CJM-1997-022-4