Advertisement

On congruences with products of variables from short intervals and applications

  • Jean Bourgain
  • Moubariz Z. Garaev
  • Sergei V. Konyagin
  • Igor E. Shparlinski
Article

Abstract

We obtain upper bounds on the number of solutions to congruences of the type (x 1 + s)... (x ν + s) ≡ (y 1 + s)... (x ν + s) ≢ 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.

Keywords

STEKLOV Institute Algebraic Number Independent Vector Algebraic Integer Implied Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Ayyad, T. Cochrane, and Z. Zheng, “The Congruence x 1 x 2x 3 x 4 (mod p), the Equation x 1 x 2 = x 3 x 4, and Mean Values of Character Sums,” J. Number Theory 59, 398–413 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    U. Betke, M. Henk, and J. M. Wills, “Successive-Minima-Type Inequalities,” Discrete Comput. Geom. 9, 165–175 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Bourgain, “More on the Sum-Product Phenomenon in Prime Fields and Its Applications,” Int. J. Number Theory 1, 1–32 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski, “On the Hidden Shifted Power Problem,” SIAM J. Comput. 41(6), 1524–1557 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    T. H. Chan and I. E. Shparlinski, “On the Concentration of Points on Modular Hyperbolas and Exponential Curves,” Acta Arith. 142, 59–66 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M.-C. Chang, “Factorization in Generalized Arithmetic Progressions and Applications to the Erdős-Szemerédi Sum-Product Problems,” Geom. Funct. Anal. 13, 720–736 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M.-C. Chang, “Character Sums in Finite Fields,” in Finite Fields: Theory and Applications (Am. Math. Soc., Providence, RI, 2010), pp. 83–98.CrossRefGoogle Scholar
  8. 8.
    J. Cilleruelo and M. Z. Garaev, “Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications,” Geom. Funct. Anal. 21, 892–904 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    T. Cochrane and S. Shi, “The Congruence x 1 x 2x 3 x 4 (mod m) and Mean Values of Character Sums,” J. Number Theory 130, 767–785 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    T. Cochrane and Z. Zheng, “High Order Moments of Character Sums,” Proc. Am. Math. Soc. 126, 951–956 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J. B. Friedlander and H. Iwaniec, “The Divisor Problem for Arithmetic Progressions,” Acta Arith. 45, 273–277 (1985).MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. B. Friedlander and H. Iwaniec, “Estimates for Character Sums,” Proc. Am. Math. Soc. 119, 365–372 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Z. Garaev, “On Multiplicative Congruences,” Math. Z. 272, 473–482 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Z. Garaev and V. C. Garcia, “The Equation x 1 x 2 = x 3 x 4 + λ in Fields of Prime Order and Applications,” J. Number Theory 128, 2520–2537 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. A. Karatsuba, “On a Certain Arithmetic Sum,” Dokl. Akad. Nauk SSSR 199(4), 770–772 (1971) [Sov. Math., Dokl. 12, 1172–1174 (1971)].Google Scholar
  16. 16.
    A. A. Karatsuba, “Weighted Character Sums,” Izv. Ross. Akad. Nauk, Ser. Mat. 64(2), 29–42 (2000) [Izv. Math. 64, 249–263 (2000)].MathSciNetGoogle Scholar
  17. 17.
    A. A. Karatsuba, “Arithmetic Problems in the Theory of Dirichlet Characters,” Usp. Mat. Nauk 63(4), 43–92 (2008) [Russ. Math. Surv. 63, 641–690 (2008)].MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. V. Konyagin, “Estimates of Character Sums in Finite Fields,” Mat. Zametki 88(4), 529–542 (2010) [Math. Notes 88, 503–515 (2010)].MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Le Boudec, “Power-Free Values of the Polynomial t 1 ... t r−1,” Bull. Aust. Math. Soc. 85(1), 154–163 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Mignotte, Mathematics for Computer Algebra (Springer, New York, 1992).CrossRefzbMATHGoogle Scholar
  21. 21.
    W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers (Pol. Sci. Publ., Warszawa, 1990).zbMATHGoogle Scholar
  22. 22.
    I. E. Shparlinski, “On the Distribution of Points on Multidimensional Modular Hyperbolas,” Proc. Jpn. Acad. A 83(2), 5–9 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    I. E. Shparlinski, “On a Generalisation of a Lehmer Problem,” Math. Z. 263, 619–631 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    T. Tao and V.H. Vu, Additive Combinatorics (Cambridge Univ. Press, Cambridge, 2006), Cambridge Stud. Adv. Math. 105.CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Jean Bourgain
    • 1
  • Moubariz Z. Garaev
    • 2
  • Sergei V. Konyagin
    • 3
  • Igor E. Shparlinski
    • 4
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Centro de Ciencias MatemáticasUniversidad Nacional Autónoma de MéxicoMorelia, MichoacánMéxico
  3. 3.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  4. 4.Department of ComputingMacquarie UniversitySydneyAustralia

Personalised recommendations