On minimum norm of representatives of residue classes in number fields. (English) Zbl 1139.11035

Let \(K\) be an algebraic number field, let \(O = O_K\) be the ring of integers of \(K\), and let \(I\) be an integral ideal of \(K\). Define
\[ L(K, I) = \max_{\alpha \in (O/I)^*} \min_{x \in \alpha} | N_{K/\mathbb Q}(x)| . \]
The inequality \(L(K, I) < N(I) = | O/I| \) for all integral ideals \(I\) means that \(K\) is Euclidean with respect to its field norm.
In this paper, the authors establish bounds for \(L(K, I)\) when \(N(I)\) is large. In particular, they show that if \(K\) has an infinite group of units (i.e., \(K\) is not an imaginary quadratic field), then the bound \(L(K, I) = o(N(I))\) holds for almost all integral ideals \(I\) in \(O\). This answers in the affirmative a question posed by S. V. Konyagin and I. E. Shparlinski [Character sums with exponential functions and their applications. Cambridge: Cambridge University Press (1999; Zbl 0933.11001)].
In the case of prime ideals, the authors use their recent bounds on exponential sums over multiplicative groups to obtain substantially stronger results. Suppose again that \(K\) has an infinite group of units. The authors show that for each \(\varepsilon > 0\), there exists a \(\delta = \delta(\varepsilon, K) > 0\) such that
\[ \# \left\{ P : P \text{ a prime ideal of } O,\;N(P)\leq T,\;L(K, P) > N(P)^{1 - \delta} \right\} < T^{\varepsilon} \]
for all sufficiently large \(T\).


11L07 Estimates on exponential sums
11L05 Gauss and Kloosterman sums; generalizations
11R04 Algebraic numbers; rings of algebraic integers
11R27 Units and factorization


Zbl 0933.11001
Full Text: DOI


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