Duke Mathematical Journal

On the minimum norm of representatives of residue classes in number fields

Jean Bourgain and Mei-Chu Chang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we consider the problem of finding upper bounds on the minimum norm of representatives in residue classes in quotient O/I, where I is an integral ideal in the maximal order O of a number field K. In particular, we answer affirmatively a question of Konyagin and Shparlinski [KS], stating that an upper bound o(N(I)) holds for most ideals I, denoting N(I) the norm of I. More precise statements are obtained, especially when I is prime. We use the method of exponential sums over multiplicative groups, essentially exploiting some new bounds obtained by the authors

Article information

Source
Duke Math. J., Volume 138, Number 2 (2007), 263-280.

Dates
First available in Project Euclid: 5 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1181051032

Digital Object Identifier
doi:10.1215/S0012-7094-07-13824-9

Mathematical Reviews number (MathSciNet)
MR2318285

Zentralblatt MATH identifier
1139.11035

Subjects
Primary: 11L051 11R27: Units and factorization
Secondary: 11L07: Estimates on exponential sums 11R04: Algebraic numbers; rings of algebraic integers

Citation

Bourgain, Jean; Chang, Mei-Chu. On the minimum norm of representatives of residue classes in number fields. Duke Math. J. 138 (2007), no. 2, 263--280. doi:10.1215/S0012-7094-07-13824-9. https://projecteuclid.org/euclid.dmj/1181051032


Export citation

References

  • D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), 509--532.
  • J. Bourgain, Mordell's exponential sum estimate revisited, J. Amer. Math. Soc. 18 (2005), 477--499.
  • —, A remark on quantum ergodicity for cat maps, to appear in the proceedings of the Geometric and Functional Analysis Seminar 2005, preprint, 2005.
  • J. Bourgain and M.-C. Chang, A Gauss sum estimate in arbitrary finite fields, C. R. Math. Acad. Sci. Paris 342 (2006), 643--646.
  • —, Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z_Q^*$, where $Q$ is composite with few prime factors, Geom. Funct. Anal. 16 (2006), 327--366.
  • J. Bourgain, A. A. Glibichuk, and S. V. Konyagin, Estimate 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.
  • S. Egami, The distribution of residue classes modulo $\fraka$ in an algebraic number field, Tsukuba J. Math. 4 (1980), 9--13.
  • H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1--49.
  • H. Halberstam and K. F. Roth, Sequences, Vol. I, Clarendon Press, Oxford, 1966.
  • S. V. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and Their Applications, Cambridge Tracts in Math. 136, Cambridge Univ. Press, Cambridge, 1999.
  • P. Kurlberg and C. Pomerance, On the periods of the linear congruential and power generators, Acta Arith. 119 (2005), 149--169.
  • P. Kurlberg and Z. Rudnick, On quantum ergodicity for linear maps of the torus, Comm. Math. Phys. 222 (2001), 201--227.
  • E. Lindenstrauss, personal communication, 2005.
  • W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Monogr. Matematyczne 57, Polish Sci., Warsaw, 1974.