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Divisibility problems for function fields

  • S. Baier
  • A. Bansal
  • R. Kumar Singh
Article
  • 10 Downloads

Abstract

We investigate three combinatorial problems considered by Erdős, Rivat, Sárközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.

Key words and phrases

divisibility problem function field large sieve 

Mathematics Subject Classification

11B75 11N36 11L40 11P99 

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Notes

Acknowledgement

We thank Igor Shparlinski for making us aware of Theorems 1.2 and 1.3 and suggesting to consider them in the function field setting.

References

  1. 1.
    S. Baier, A note on P-sets, Integers, 4 (2004), A13, 6 pp.Google Scholar
  2. 2.
    Baier S.: On the large sieve with sparse sets of moduli. J. Ramanujan Math. Soc., 21, 279–295 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Baier and R. K. Singh, Large sieve inequality with power moduli for function fields, arXiv:1802.03131 [math.NT] (2018).
  4. 4.
    Baier S., ZhaoL L.: Large sieve inequality with characters for powerful moduli. Int. J. Number Theory, 1, 265–279 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    E. Bombieri, Le grand crible dans la thorie analytique des nombres, 2nd ed., Astérisque, 18, Société Mathématique de France (Paris, 1987).Google Scholar
  6. 6.
    Elsholtz C., Planitzer S.: On Erdős and Sárközy’s sequences with Property P. Monatsh. Math., 182, 565–575 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős P., Sárközy A.: On divisibility properties of integers of the form aa. Acta Math. Hungar., 50, 117–122 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hsu C.-N.: A large sieve inequality for rational function fields. J. Number Theory, 58, 267–287 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. L. Montgomery, Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag (Berlin–New York, 1971).Google Scholar
  10. 10.
    J. Rivat and A. Sárközy, On arithmetic properties of products and shifted products, in: Analytic Number Theory, Springer (Cham, 2015), pp. 345–355.Google Scholar
  11. 11.
    Schoen T.: On a problem of Erdős and Sárközy. J. Comb. Theory Ser. A, 94, 191–195 (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Zhao L.: Large sieve inequality with characters to square moduli. Acta Arith., 112, 297–308 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of MathematicsRamakrishna Mission Vivekananda Educational Research InstituteHowrahIndia
  2. 2.School of Physical SciencesJawaharlal Nehru UniversityNew-DelhiIndia

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