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Fine Spectra and Limit Laws I. First-Order Laws

Published online by Cambridge University Press:  20 November 2018

Stanley Burris  and
András Sárközy
Stanley Burris
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1
András Sárközy
Affiliation:
Department of Mathematics, University L. Eötvös, Department of Algebra and Number Theory, H–1088 Budapest, Muzeum krt. 6–8, Hungary
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Abstract

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Using Feferman-Vaught techniques we show a certain property of the fine spectrumof an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does 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.

Keywords

O3C1311N4511N8005A1505A1611M4111P81First order limit lawsgeneralized number theory
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 3 , 01 June 1997 , pp. 468 - 498
Copyright
Copyright © Canadian Mathematical Society 1997

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