## Duke Mathematical Journal

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

#### 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

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

#### 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

#### 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.