On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one

Stephan Baier; Marc Technau

doi:10.5802/jtnb.1141

On the distribution of

α p

modulo one in imaginary quadratic number fields with class number one

Stephan Baier ¹ ; Marc Technau ²

¹ Ramakrishna Mission Vivekananda Educational Research Institute Department of Mathematics G. T. Road, PO Belur Math, Howrah West Bengal 711202, India
² Graz University of Technology Institute of Analysis and Number Theory Kopernikusgasse 24/II 8010 Graz, Austria

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 719-760.

Résumé
Abstract

Nous étudions la répartition de $α p$ modulo un dans les corps quadratiques imaginaires $𝕂 \subset ℂ$ dont le nombre de classes est égal à un, où $p$ parcourt l’ensemble des idéaux premiers de l’anneau des entiers $𝒪 = ℤ [ω]$ de $𝕂$ . Par analogie avec un résultat classique dû à R. C. Vaughan, nous obtenons que l’inégalité ${∥ α p ∥}_{ω} < N {(p)}^{- 1 / 8 + ϵ}$ est satisfaite pour une infinité de $p$ , où ${∥ ϱ ∥}_{ω}$ mesure la distance de $ϱ \in ℂ$ à $𝒪$ et $N (p)$ est la norme de $p$ .

La preuve est basée sur la méthode du crible de Harman et utilise des analogues pour les corps de nombres d’idées classiques dues à Vinogradov. De plus, nous introduisons un lissage qui nous permet d’utiliser la formule sommatoire de Poisson.

We investigate the distribution of $α p$ modulo one in imaginary quadratic number fields $𝕂 \subset ℂ$ with class number one, where $p$ is restricted to prime elements in the ring of integers $𝒪 = ℤ [ω]$ of $𝕂$ . In analogy to classical work due to R. C. Vaughan, we obtain that the inequality ${∥ α p ∥}_{ω} < N {(p)}^{- 1 / 8 + ϵ}$ is satisfied for infinitely many $p$ , where ${∥ ϱ ∥}_{ω}$ measures the distance of $ϱ \in ℂ$ to $𝒪$ and $N (p)$ denotes the norm of $p$ .

The proof is based on Harman’s sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Reçu le : 2019-12-02
Révisé le : 2020-05-19
Accepté le : 2020-09-18
Publié le : 2021-01-08

DOI : 10.5802/jtnb.1141

Classification : 11J17, 11L07, 11L20, 11K60
Mots clés : Distribution modulo one, Diophantine approximation, imaginary quadratic field, smoothed sum, Poisson summation

Affiliations des auteurs :

Stephan Baier ¹ ; Marc Technau ²

¹ Ramakrishna Mission Vivekananda Educational Research Institute Department of Mathematics G. T. Road, PO Belur Math, Howrah West Bengal 711202, India
² Graz University of Technology Institute of Analysis and Number Theory Kopernikusgasse 24/II 8010 Graz, Austria

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2020__32_3_719_0,
     author = {Stephan Baier and Marc Technau},
     title = {On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {719--760},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {3},
     year = {2020},
     doi = {10.5802/jtnb.1141},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1141/}
}

TY  - JOUR
AU  - Stephan Baier
AU  - Marc Technau
TI  - On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 719
EP  - 760
VL  - 32
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1141/
DO  - 10.5802/jtnb.1141
LA  - en
ID  - JTNB_2020__32_3_719_0
ER  -

%0 Journal Article
%A Stephan Baier
%A Marc Technau
%T On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 719-760
%V 32
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1141/
%R 10.5802/jtnb.1141
%G en
%F JTNB_2020__32_3_719_0

Stephan Baier; Marc Technau. On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 719-760. doi : 10.5802/jtnb.1141. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1141/

Bibliographie
Cité par

[1] Stephan Baier A note on Diophantine approximation with Gaussian primes (2016) (https://arxiv.org/abs/1609.08745)

[2] Alan Baker Linear forms in the logarithms of algebraic numbers, Mathematika, Volume 13 (1966), pp. 204-216 | DOI | MR | Zbl

[3] Jörg Brüdern Einführung in die analytische Zahlentheorie, Springer, 1995 | Zbl

[4] H. Gintner Über Kettenbruchentwicklung und über die Approximation von komplexen Zahlen, Ph. D. Thesis, University of Vienna (1936)

[5] Godfrey H. Hardy; Edward M. Wright An introduction to the theory of numbers, Oxford University Press, 2008

[6] Glyn Harman On the distribution of $α p$ modulo one, J. Lond. Math. Soc., Volume 27 (1983) no. 1, pp. 9-18 | DOI | MR

[7] Glyn Harman On the distribution of $α p$ modulo one. II, Proc. Lond. Math. Soc., Volume 72 (1996) no. 2, pp. 241-260 | DOI | MR | Zbl

[8] Glyn Harman Prime-detecting sieves, London Mathematical Society Monographs, 33, Princeton University Press, 2007 | MR | Zbl

[9] Glyn Harman Diophantine approximation with Gaussian primes (2019) (Preprint, to appear) | DOI

[10] David R. Heath-Brown; Chaohua Jia The distribution of $α p$ modulo one, Proc. Lond. Math. Soc., Volume 84 (2002) no. 1, pp. 79-104 | DOI | MR | Zbl

[11] Kurt Heegner Diophantische Analysis und Modulfunktionen, Math. Z., Volume 56 (1952), pp. 227-253 | DOI | MR | Zbl

[12] Chaohua Jia On the distribution of $α p$ modulo one, J. Number Theory, Volume 45 (1993) no. 3, pp. 241-253 | Zbl

[13] Chaohua Jia On the distribution of $α p$ modulo one. II, Sci. China, Ser. A, Volume 43 (2000) no. 7, pp. 703-721 | MR | Zbl

[14] Guangshi Lü On mean values of some arithmetic functions in number fields, Acta Math. Hung., Volume 132 (2011) no. 4, pp. 348-357 | MR | Zbl

[15] Kaisa Matomäki The distribution of $α p$ modulo one, Math. Proc. Camb. Philos. Soc., Volume 147 (2009) no. 2, pp. 267-283 | DOI | MR | Zbl

[16] Harold M. Stark A complete determination of the complex quadratic fields of class-number one, Mich. Math. J., Volume 14 (1967), pp. 1-27 | MR | Zbl

[17] Harold M. Stark On the ‘gap’ in a theorem of Heegner, J. Number Theory, Volume 1 (1969), pp. 16-27 | DOI | MR | Zbl

[18] Elias M. Stein; Guido Weiss Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, Princeton University Press, 1971 | Zbl

[19] Marc Technau On Beatty sets and some generalisations thereof, Ph. D. Thesis, University of Würzburg (2018) | Zbl

[20] Robert C. Vaughan On the distribution of $α p$ modulo $1$ , Mathematika, Volume 24 (1978), pp. 135-141 | DOI

[21] Ivan M. Vinogradov The method of trigonometrical sums in the theory of numbers, Dover Publications, 2004 (Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport. Reprint of the 1954 translation.)

Cité par Sources :