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# Fine spectra and limit laws I. First-order laws

Published:1997-06-01
Printed: Jun 1997
• Stanley Burris
• András Sárközy
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## Abstract

Using Feferman-Vaught techniques we show a certain property of the fine spectrum of 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, {\it 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: First order limit laws, generalized number theory
 MSC Classifications: O3C13 - unknown classification O3C1311N45 - Asymptotic results on counting functions for algebraic and topological structures 11N80 - Generalized primes and integers 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16 - Asymptotic enumeration 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11P81 - Elementary theory of partitions [See also 05A17]

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