Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-22T20:45:01.769Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Bateman, P.T. and Diamond, H.G., Asymptotic distribution of Beurling's generalized prime numbers, . In: Studies in Number Theory, (ed. LeVeque, W.), MAA Stud. Math. 6, Prentice-Hall, 1969.Google Scholar
2. Beurling, A., Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I, Acta Math. 68(1937), 255–291.Google Scholar
3. Bigelow, D. and Burris, S., Boolean algebras of factor congruences, Acta Sci. Math. (Szeged) 54(1990), 11–20.Google Scholar
4. Burris, S. and Idziak, P., A directly representable variety has a discrete first-order law, International J. Algebra Comput. 6(1996), 269–276.Google Scholar
5. Compton, K., A logical approach to asymptotic combinatorics. I. First order properties, Adv. in Math. 65(1987), 65–96.Google Scholar
6. Compton, K.J., Henson, C.W. and Shelah, S., Nonconvergence, undecidability, and intractability in asymptotic problems, Ann. Pure Appl. Logic 36(1987), 207–224.Google Scholar
7. Compton, K., Odlyzko, A. and Richmond, B., Fine spectra and limit laws II. First-order0–1 laws.Google Scholar
8. Feferman, S. and Vaught, R.L., The first-order properties of algebraic systems, Fund. Math. 47(1959), 57–103.Google Scholar
9. Geluk, J.L. and de Haan, L., Regular variation, extensions and Tauberian theorems, CWI Tract 40, Centre for Mathematics and Computer Science, 1987.Google Scholar
10. Higman, G., Enumerating p-groups. I: Inequalities, Proc. London Math. Soc. 10(1960), 24–30.Google Scholar
11. Knopfmacher, J., Arithmetical properties of finite rings and algebras, and analytic number theory. I, J. Reine Angew. Math. 252(1972), 16–43.Google Scholar
12. Knopfmacher, J., Arithmetical properties of finite rings and algebras, and analytic number theory. III. Finite modules and algebras over Dedekind domains, J. Reine Angew. Math. 259(1973), 157–170.Google Scholar
13. Knopfmacher, J., Abstract Analytic Number Theory, North Holland, 1975.Google Scholar
14. Knopfmacher, J., Analytic Arithmetic of Algebraic Function Fields, Marcel Dekker, Inc., 1979.Google Scholar
15. Knopfmacher, A., Knopfmacher, J. and Warlimont, R., “Factorisatio numerorum” in arithmetical semigroups, Acta Arith. 61(1992), 328–336.Google Scholar
16. Landau, E., Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes, Math. Ann. 56(1903), 645–670.Google Scholar
17. McKenzie, R., McNulty, G., and Taylor, W., Algebras, Varieties, Lattices. Vol. I, Wadsworth and Cole/Brooks, 1987.Google Scholar
18. Oppenheim, A., On an arithmetic function, J. London Math. Soc. 1(1926), 105–211.Google Scholar
19. Oppenheim, A., On an arithmetic function (II), J. London Math. Soc. 2(1927), 123–130.Google Scholar
20. Skolem, Th., Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, Videnskabsakademiet i Kristiania, Skrifter I, 3, 1919. 37.Google Scholar
21. Szekeres, G. and Turán, P., Über das zweite Hauptproblem der “Factorisatio Numerorum”, Acta Litt. Szeged 6(1933), 143–154.Google Scholar
22. Taylor, W., The fine spectrum of a variety, Algebra Universalis 5(1975), 263–303.Google Scholar
23. Titchmarsh, E.C., The theory of functions, 2nd ed., Oxford Univ. Press, 1939.Google Scholar
24. Wilf, H., Generating functionology, 2nd ed., Academic Press, 1994.Google Scholar
25. Willard, R., Varieties having Boolean factor congruences, J. Algebra 130(1990), 130–153.Google Scholar