×

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

Citations:

Zbl 0933.11001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. Berend, Multi-invariant sets on tori , Trans. Amer. Math. Soc. 280 (1983), 509–532. JSTOR: · Zbl 0532.10028 · doi:10.2307/1999631
[2] J. Bourgain, Mordell’s exponential sum estimate revisited , J. Amer. Math. Soc. 18 (2005), 477–499. · Zbl 1072.11063 · doi:10.1090/S0894-0347-05-00476-5
[3] -, A remark on quantum ergodicity for cat maps , to appear in the proceedings of the Geometric and Functional Analysis Seminar 2005, preprint, 2005.
[4] J. Bourgain and M.-C. Chang, A Gauss sum estimate in arbitrary finite fields , C. R. Math. Acad. Sci. Paris 342 (2006), 643–646. · Zbl 1127.11081 · doi:10.1016/j.crma.2006.01.022
[5] -, Exponential sum estimates over subgroups and almost subgroups of \(\mathbb Z_Q^*\), where \(Q\) is composite with few prime factors , Geom. Funct. Anal. 16 (2006), 327–366. · Zbl 1183.11047 · doi:10.1007/s00039-006-0558-7
[6] J. Bourgain, A. A. Glibichuk, and S. V. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order , J. London Math. Soc. (2) 73 (2006), 380–398. · Zbl 1093.11057 · doi:10.1112/S0024610706022721
[7] S. Egami, The distribution of residue classes modulo \(\fraka\) in an algebraic number field , Tsukuba J. Math. 4 (1980), 9–13. · Zbl 0455.12009
[8] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation , Math. Systems Theory 1 (1967), 1–49. · Zbl 0146.28502 · doi:10.1007/BF01692494
[9] H. Halberstam and K. F. Roth, Sequences, Vol. I , Clarendon Press, Oxford, 1966.
[10] S. V. Konyagin and I. E. Shparlinski, Character Sums with Exponential Functions and Their Applications , Cambridge Tracts in Math. 136 , Cambridge Univ. Press, Cambridge, 1999. · Zbl 0933.11001
[11] P. Kurlberg and C. Pomerance, On the periods of the linear congruential and power generators , Acta Arith. 119 (2005), 149–169. · Zbl 1080.11059 · doi:10.4064/aa119-2-2
[12] P. Kurlberg and Z. Rudnick, On quantum ergodicity for linear maps of the torus , Comm. Math. Phys. 222 (2001), 201–227. · Zbl 1042.81026 · doi:10.1007/s002200100501
[13] E. Lindenstrauss, personal communication, 2005.
[14] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers , Monogr. Matematyczne 57 , Polish Sci., Warsaw, 1974. · Zbl 0276.12002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.