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Group Topologies on the Integers and S-Unit Equations

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Abstract

A sequence of integers is called a T-sequence if there exists a Hausdorff group topology on the integers such that the sequence converges to 0. Given a finite set S of primes, we construct some Hausdorff group topology on the integers such that every increasing sequence with terms divisible only by primes from S converges to 0. Also we answer in the affirmative the question on T-sequences which was posed by Protasov and Zelenuk. Our results rely on a nontrivial number-theoretic fact about S-unit equations.

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References

  1. Zelenyuk E. G. and Protasov I. V., “Topologies on Abelian groups,” Math. USSR-Izv., vol. 37, no. 2, 445–460 (1991).

    Article  MathSciNet  Google Scholar 

  2. Protasov I. V. and Zelenuk E. G., “Topologies on ℤ determined by sequences: seven open problems,” Mat. Stud., vol. 12, no. 1, 111 (1999).

    MathSciNet  MATH  Google Scholar 

  3. Mazurov V. D. and Khukhro E. I. (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 19th ed., Sobolev Inst. Math., Novosibirsk (2018).

    MATH  Google Scholar 

  4. Maynard J., “Small gaps between primes,” Ann. Math. (2), vol. 181, no. 1, 383–413 (2015).

    Article  MathSciNet  Google Scholar 

  5. Protasov I. and Zelenyuk E., Topologies on Groups Determined by Sequences, VNTL Publ., Lviv (1999) (Math. Stud. Monogr. Ser.; V. 4).

    MATH  Google Scholar 

  6. Higasikawa M., “Non-productive duality properties of topological groups,” Topol. Proc., vol. 25, 207–216 (2000).

    MathSciNet  MATH  Google Scholar 

  7. Evertse J.-H. and Győry K., Unit Equations in Diophantine Number Theory, Cambridge Univ. Press, Cambridge (2015) (Camb. Stud. Adv. Math.).

    Book  Google Scholar 

  8. Ruzsa I., “Arithmetical topology,” in: Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. Janos Bolyai, 51, North-Holland, Amsterdam, 1990, 473–504.

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    S. V. Skresanov

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  1. S. V. Skresanov
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Correspondence to S. V. Skresanov.

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Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 3, pp. 687–691.

The work was supported by the Program of Fundamental Scientific Research of the Russian Federation (Project 0314-2019-0001).

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Skresanov, S.V. Group Topologies on the Integers and S-Unit Equations. Sib Math J 61, 542–544 (2020). https://doi.org/10.1134/S0037446620030179

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  • DOI: https://doi.org/10.1134/S0037446620030179

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