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

Baier, Stephan; Technau, Marc

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

Nous étudions la répartition de $\alpha p$ 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 $𝒪=\mathbb{Z}\left[\omega \right]$ de $𝕂$. Par analogie avec un résultat classique dû à R. C. Vaughan, nous obtenons que l’inégalité ${\parallel \alpha p\parallel }_{\omega }<\mathrm{N}{\left(p\right)}^{-1/8+ϵ}$ est satisfaite pour une infinité de $p$, où ${\parallel \varrho \parallel }_{\omega }$ mesure la distance de $\varrho \in ℂ$ à $𝒪$ et $\mathrm{N}\left(p\right)$ 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 $\alpha p$ 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 $𝒪=\mathbb{Z}\left[\omega \right]$ of $𝕂$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality ${\parallel \alpha p\parallel }_{\omega }<\mathrm{N}{\left(p\right)}^{-1/8+ϵ}$ is satisfied for infinitely many $p$, where ${\parallel \varrho \parallel }_{\omega }$ measures the distance of $\varrho \in ℂ$ to $𝒪$ and $\mathrm{N}\left(p\right)$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[21] Vinogradov, Ivan M. 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.)