Duke Mathematical Journal
- Duke Math. J.
- Volume 138, Number 2 (2007), 263-280.
On the minimum norm of representatives of residue classes in number fields
In this article, we consider the problem of finding upper bounds on the minimum norm of representatives in residue classes in quotient , where is an integral ideal in the maximal order of a number field . In particular, we answer affirmatively a question of Konyagin and Shparlinski [KS], stating that an upper bound holds for most ideals , denoting the norm of . More precise statements are obtained, especially when is prime. We use the method of exponential sums over multiplicative groups, essentially exploiting some new bounds obtained by the authors
Duke Math. J., Volume 138, Number 2 (2007), 263-280.
First available in Project Euclid: 5 June 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11L051 11R27: Units and factorization
Secondary: 11L07: Estimates on exponential sums 11R04: Algebraic numbers; rings of algebraic integers
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