Skip to main content
SpringerLink
Log in

Additive Complements for a Given Asymptotic Density

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We investigate the existence of subsets A and B of \({\mathbb {N}}:=\{0,1,2,\dots \}\), such that the sumset \(A+B:=\{a+b:a\in A,b\in B\}\) has prescribed asymptotic density. We solve the particular case in which B is a given finite subset of \({\mathbb {N}}\) and also the case when \(B=A\); in the later case, we generalize our result to \(kA:=\{x_1+\cdots +x_k: x_i\in A, i=1,\dots ,k\}\) for an integer \(k\ge 2.\)

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.

Similar content being viewed by others

References

  1. Bienvenu, P.-Y., Hennecart, F.: On the density or measure of sets and their sumsets in the integers and on the circle. J. Number Theory 212, 285–310 (2020)

    Article  MathSciNet  Google Scholar 

  2. Cassels, J.W.S.: Über Basen der natürlichen Zahlenreihe. Abh. Math. Semin. Univ. Hamb. 21, 247–257 (1957)

    Article  Google Scholar 

  3. Cheo, L.P.: A remark on the \(\alpha +\beta \)-theorem. Proc. Am. Math. Soc. 3, 175–177 (1952)

    MathSciNet  MATH  Google Scholar 

  4. Chen, Y.-G., Fang, J.-H.: On a conjecture of Sárközy and Szemerédi. Acta Arith. 169, 47–58 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chen, Y.-G., Fang, J.-H.: On a conjecture of additive complements. Q. J. Math. 70, 927–936 (2019)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y.-G., Fang, J.-H.: Additive complements with Narkiewicz’s condition. Combinatorica 39, 813–823 (2019)

    Article  MathSciNet  Google Scholar 

  7. Danzer, L.: Über eine Frage von G. Hanani aus der additiven Zahlentheorie. J. Reine Angew. Math. 214/215, 392–394 (1964)

    MATH  Google Scholar 

  8. Fang, J.-H., Chen, Y.-G.: On additive complements III. J. Number Theory 141, 83–91 (2014)

    Article  MathSciNet  Google Scholar 

  9. Freedman, A.R., Sember, J.J.: Densities and summability. Pac. J. Math. 95, 293–305 (1981)

    Article  MathSciNet  Google Scholar 

  10. Grekos, G.: On various definitions of density (survey). Tatra Mt. Math. Publ. 31, 17–27 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Hegyvári, N., Hennecart, F., Pach, P.P.: On the density of sumsets and product sets. Australas. J. Combin. 74, 1–16 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Halberstam, H., Roth, K.F.: Sequences, volume I, Oxford 1966, 2nd edn. Springer, Berlin (1983)

    Google Scholar 

  13. Kneser, M.: Abschätzung der asymptotischen Dichte von Summenmengen. Math. Z. 58, 459–484 (1953)

    Article  MathSciNet  Google Scholar 

  14. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover, New York (2005)

    MATH  Google Scholar 

  15. Leonetti, P., Tringali, S.: On the notions of upper and lower density. Proc. Edinb. Math. Soc. 63(1), 139–167 (2020)

    Article  MathSciNet  Google Scholar 

  16. Leonetti, P., Tringali, S.: Upper and lower densities have the strong Darboux property. J. Number Theory 174, 445–455 (2017)

    Article  MathSciNet  Google Scholar 

  17. Leonetti, P., Tringali, S.: On small sets of integers. arXiv: 1905.08075. Ramanujan J. (2021) (To appear)

  18. Leonetti, P., Tringali, S.: On the density of sumsets. arXiv: 2001.10413 (2020)

  19. Lepson, B.: Certain best possible results in the theory of Schnirelmann density. Proc. Am. Math. Soc. 1, 592–594 (1950)

    Article  MathSciNet  Google Scholar 

  20. Lorentz, G.G.: On a problem of additive number theory. Proc. Am. Math. Soc. 5, 838–841 (1954)

    Article  Google Scholar 

  21. Narkiewicz, W.: Remarks on a conjecture of Hanani in additive Number Theory. Colloq. Math. 7, 161–165 (1959/1960)

  22. Nathanson, M.B.: Best possible results on the density of subsets. In: “Analytic Number Theory”, Proceedings of Conference in honor of Paul T. Bateman, Urbana, IL (USA), Progress in Mathematics, 85, vol. 1990, pp. 395–403. Birkhäuser (1989)

  23. Ruzsa, I.Z.: Exact additive complements. Q. J. Math. 68, 227–235 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Strauch, O., Porubský, Š.: Distribution of Sequences: A Sampler, Peter Lang, corrected version on-line. https://math.boku.ac.at/udt/ (2005)

  25. Volkmann, B.: On uniform distribution and the density of sum sets. Proc. Am. Math. Soc. 8, 130–136 (1957)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Salvatore Tringali for having suggested the idea used in the proof of Case A of Theorem 2.3; to Władysław Narkiewicz, Joël Rivat, Andrzej Schinzel and Bodo Volkmann for fruitful discussions; to the anonymous referee for suggestions and remarks, that led us to render the text of the previous version richer and more precise. The late Professor Hédi Daboussi was present and made useful observations on this work in the seminar “Rencontres de théorie analytique et élémentaire des nombres” (Paris, December 2018).

Author information

Authors and Affiliations

  1. Institut Camille Jordan, Université de Saint-Étienne, 42023, Saint-Étienne Cedex 2, France

    Alain Faisant & Georges Grekos

  2. Department of Mathematics, Indian Institute of Technology, Roorkee, 247667, India

    Ram Krishna Pandey

  3. Aspireal Technologies Private Limited, Jubilee Hills, Hyderabad, Telangana,  500033, India

    Sai Teja Somu

Authors
  1. Alain Faisant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georges Grekos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ram Krishna Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sai Teja Somu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georges Grekos.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faisant, A., Grekos, G., Pandey, R.K. et al. Additive Complements for a Given Asymptotic Density. Mediterr. J. Math. 18, 25 (2021). https://doi.org/10.1007/s00009-020-01679-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-01679-0

Keywords

Mathematics Subject Classification

Search

Navigation