Abstract view
Fine spectra and limit laws I. Firstorder laws


Published:19970601
Printed: Jun 1997
Stanley Burris
András Sárközy
Abstract
Using FefermanVaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a firstorder 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 firstorder 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.
MSC Classifications: 
O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 show english descriptions
unknown classification O3C13 Asymptotic results on counting functions for algebraic and topological structures Generalized primes and integers Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] Asymptotic enumeration Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebrogeometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} Elementary theory of partitions [See also 05A17]
O3C13  unknown classification O3C13 11N45  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 algebrogeometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11P81  Elementary theory of partitions [See also 05A17]
