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On conjectures and problems of Ruzsa concerning difference graphs of S-units

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Abstract

Given a finite nonempty set of primes S, we build a graph \({\mathcal{G}}\) with vertex set \({\mathbb{Q}}\) by connecting \({x, y \in \mathbb{Q}}\) if the prime divisors of both the numerator and denominator of x − y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of \({\mathcal{G}}\), and also a problem of Ruzsa concerning the existence of subgraphs of \({\mathcal{G}}\) which are not induced subgraphs.

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References

  1. J. H. Coates, An effective p-adic analogue of a theorem of Thue, Acta Arith., 15 (1969), 279–305.

  2. J.-H. Evertse, On sums of S-units and linear recurrences, Comp. Math., 53 (1984), 225–244.

  3. Evertse J.-H.: On equations in S-units and the Thue–Mahler equation. Invent. Math. 78, 561–584 (1984)

    Article  MathSciNet  Google Scholar 

  4. K. Győry, On certain graphs associated with an integral domain and their applications to Diophantine problems, Publ. Math. Debrecen, 29 (1982), 79–94.

  5. K. Győry, On certain arithmetic graphs and their applications to Diophantine problems, Func. Approx. Comment. Math., 39 (2008), 289–314.

  6. K. Győry, L. Hajdu and R. Tijdeman, Representation of finite graphs as difference graphs of S-units, I, J. Combin. Theory Ser. A, 127 (2014), 314–335.

  7. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press (Oxford, 2008).

  8. Mahler K.: Zur Approximation algebraischer Zahlen, I, Math. Ann., 107, 691–730 (1933)

    Article  MathSciNet  Google Scholar 

  9. I. Z. Ruzsa, The difference graph of S-units, Publ. Math. Debrecen, 79 (2011), 675–685.

  10. A. J. Van der Poorten and H. P. Schlickewei, The growth condition for recurrence sequences, Macquarie Math. Reports 82–0041, North Ryde (1982). Cf. Additive relations in fields, J. Austral. Math. Soc., 51 (1991), 154–170, http://maths.mq.edu.au/~alf/www-centre/alfpapers/a092.pdf.

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Authors and Affiliations

  1. Institut für Optimierung und Diskrete Mathematik, Technische Universität Graz, Steyrergasse 30/II, 8010, Graz, Austria

    A. Ćustić

  2. Institute of Mathematics, University of Debrecen, P.O. Box 12, H-4010, Debrecen, Hungary

    L. Hajdu

  3. Institut für Analysis und Computational Number Theory, Technische Universität Graz, Steyrergasse 30/II, 8010, Graz, Austria

    D. Kreso

  4. Mathematical Institute, Leiden University, 2300 RA, P.O. Box 9512, Leiden, The Netherlands

    R. Tijdeman

Authors
  1. A. Ćustić
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  2. L. Hajdu
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  3. D. Kreso
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  4. R. Tijdeman
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Correspondence to D. Kreso.

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Ćustić, A., Hajdu, L., Kreso, D. et al. On conjectures and problems of Ruzsa concerning difference graphs of S-units. Acta Math. Hungar. 146, 391–404 (2015). https://doi.org/10.1007/s10474-015-0513-x

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  • DOI: https://doi.org/10.1007/s10474-015-0513-x

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