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