## 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 $$T$$.

### MSC:

 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:

### References:

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