Differencing methods for Korobov-type exponential sums

  • Joseph VandeheyEmail author


We study exponential sums of the form \(\Sigma_{n=1}^N\;e^{{2\pi}iab^{n}/m}\) for nonzero integers a, b, m. Classically, non-trivial bounds were known for N ≥ √m by Korobov, and this range has been extended significantly by Bourgain as a result of his and others’ work on the sum-product phenomenon. Let P be a finite set of primes and let m be a large integer whose primes factors all belong to P. We use a variant of the Weyl-van der Corput method of differencing to give more explicit bounds that become non-trivial around the time when exp(log m/ log2 logm) ≤ N. We include applications to the digits of rational numbers and constructions of normal numbers.


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The author acknowledges assistance from the Research and Training Group grant DMS-1344994 funded by the National Science Foundation.


  1. [1]
    D. H. Bailey and R. E. Crandall, Random generators and normal numbers, Experiment. Math. 11 (2002), 527–546.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. Bourgain, Exponential sum estimates over subgroups of ℤ*q, q arbitrary, J. Anal. Math. 97 (2005), 317–355.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Bourgain, The sum-product theorem in ℤ q with q arbitrary, J. Anal. Math. 106 (2008), 1–93.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), 380–398.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Bourgain, N. Katz and T. Tao, A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), 27–57.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Z. Garaev, Sums and products of sets and estimates for rational trigonometric sums in fields of prime order, Uspekhi Mat. Nauk 65 (2010), no. 4 (394), 5–66.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004.zbMATHGoogle Scholar
  8. [8]
    S.-M. Jung and B. Volkmann, Remarks on a paper of Wagner, J. Number Theory 56 (1996), 329–335.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    H. Kano and I. Shiokawa, Rings of normal and nonnormal numbers, Israel J. Math. 84 (1993), 403–416.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B. Kerr, Incomplete exponential sums over exponential functions, Q. J. Math. 66 (2015), 213–224.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. V. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and their Applications, Cambridge University Press, Cambridge, 1999.CrossRefzbMATHGoogle Scholar
  12. [12]
    N. M. Korobov, Trigonometric sums with exponential functions, and the distribution of the digits in periodic fractions, Mat. Zametki 8 (1970), 641–652.MathSciNetzbMATHGoogle Scholar
  13. [13]
    N. M. Korobov, The distribution of digits in periodic fractions, Mat. Sb. (N.S.) 89(131) (1972), 654–670, 672.MathSciNetzbMATHGoogle Scholar
  14. [14]
    N. M. Korobov, Exponential Sums and their Applications, Kluwer Academic, Dordrecht, 1992.CrossRefzbMATHGoogle Scholar
  15. [15]
    P. Kurlberg, Bounds on exponential sums over small multiplicative subgroups, in Additive Combinatorics, American Mathematical Society, Providence, RI, 2007, pp. 55–68.CrossRefGoogle Scholar
  16. [16]
    D. Milićević, Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli, Compos. Math. 152 (2016), 825–875.MathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Sándor, D. S. Mitrinović and B. Crstici, Handbook of Number Theory. I, Springer, Dordrecht, 2006zbMATHGoogle Scholar
  18. [18]
    I. Shparlinski, On exponential sums with sparse polynomials and rational functions, J. Number Theory 60 (1996), 233–244.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Wagner, On rings of normal and nonnormal numbers, preprint, no. 34, Laboratoire de Mathématiques, Marseille, 1989.Google Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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