Duke Mathematical Journal

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

Jean Bourgain and Mei-Chu Chang

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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

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

First available in Project Euclid: 5 June 2007

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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

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