Skip to main content
SpringerLink
Log in

Logarithmic density and measures on semigroups

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Davenport and Erdős [3] proved that every setA of integers with the property thata ∈A impliesan ∈A for alln (multiplicative ideal) has a logarithmic density. I generalized [5] this result to sets with the property that if for some numbersa, b, n we havea ∈ A, b ∈ A andan ∈ A, then necessarilybn ∈ A, which I call quasi-ideals.

Here a new proof of this theorem is given, applying a result on convolution of measures on discretes semigroups. This leads to further generalizations, including an improvement of a result of Warlimont [8] on ideals in abstract arithmetic semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berstel, J.; Perrin, D.,Theory of codes, Academic Press 1985

  2. Besicovitch, A. S., On the density of certain sequences of integers,Math. Annalen 110(1934), 336–341

    Article  MATH  MathSciNet  Google Scholar 

  3. Davenport, H.; Erdős, P., On sequences of positive integers,Acta Arithmetica 2(1937), 147–151

    Google Scholar 

  4. Ruzsa, I. Z., General multiplicative functions,Acta Arithmetica 32(1977), 313–347

    MATH  MathSciNet  Google Scholar 

  5. Ruzsa, I. Z., Semigroup-valued multiplicative functions,Acta Arithmetica 42(1982–83), 79–90

    MATH  MathSciNet  Google Scholar 

  6. Ruzsa, I. Z., Infinite convolution of distributions on discrete commutative semigroups, in:Probability measures on groups X. (Proc. Conf., Oberwolfach 1990), ed. H. Heyer, Plenum Press, New York (1992), 365–376

    Google Scholar 

  7. Schwarz, W.; Spilker, J.,Arithmetical functions, London Math. Soc. Lecture Note 184, Cambridge Univ. Press 1994

  8. Warlimont, R., Arithmetical semigroups II: sieving by large and small prime elements. Sets of multiples,Manuscripta Math. 71(1991), 197–221

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary

    Imre Z. Ruzsa

Authors
  1. Imre Z. Ruzsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ruzsa, I.Z. Logarithmic density and measures on semigroups. Manuscripta Math 89, 307–317 (1996). https://doi.org/10.1007/BF02567519

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567519

Keywords

Search

Navigation