Skip to main content
SpringerLink
Log in

Geometric progressions in syndetic sets

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

An open problem about finite geometric progressions in syndetic sets leads to a family of diophantine equations related to the commutativity of translation and multiplication by squares.

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. Beiglböck, M., Bergelson, V., Hindman, N., Strauss, D.: Multiplicative structures in additively large sets. J. Combin. Theory Ser. A 1137, 1219–1242 (2006)

    Article  MathSciNet  Google Scholar 

  2. Glasscock, D., Koutsogiannis, A., Richter, F.K.: Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete Contin. Dyn. Syst. 3910, 5891–5921 (2019)

    Article  MathSciNet  Google Scholar 

  3. McNew, N.: Primitive and geometric-progression-free sets without large gaps. Acta Arith. 1921, 95–104 (2020)

    Article  MathSciNet  Google Scholar 

  4. Nathanson, M.B., O’Bryant, K.: On sequences without geometric progressions. Integers 13, Paper No. A73, 5 pp. (2013)

  5. Nathanson, M.B., O’Bryant, K.: Irrational numbers associated to sequences without geometric progressions. Integers 14, Paper No. A40, 11 pp. (2014)

  6. Nathanson, M.B., O’Bryant, K.: A problem of Rankin on sets without geometric progressions. Acta Arith. 170(4), 327–342 (2015)

    Article  MathSciNet  Google Scholar 

  7. Patil, B.R.: Geometric progressions in syndetic sets. Arch. Math. (Basel) 1132, 157–168 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I wish to thank B.R. Patil for introducing me to this subject at the ICTS Workshop on Additive Combinatorics in Bangalore in March, 2020.

Author information

Authors and Affiliations

  1. Department of Mathematics, Lehman College (CUNY), Bronx, NY, 10468, USA

    Melvyn B. Nathanson

Authors
  1. Melvyn B. Nathanson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Melvyn B. Nathanson.

Additional information

Publisher's Note

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

This research was supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for the program—Workshop on Additive Combinatorics (Code: ICTS/Prog-wac2020/02).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nathanson, M.B. Geometric progressions in syndetic sets. Arch. Math. 115, 413–417 (2020). https://doi.org/10.1007/s00013-020-01488-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-020-01488-7

Keywords

Mathematics Subject Classification

Search

Navigation