On the distribution of αp modulo one in imaginary quadratic number fields with class number one
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 719-760.

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

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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1141
Classification: 11J17,  11L07,  11L20,  11K60
Keywords: Distribution modulo one, Diophantine approximation, imaginary quadratic field, smoothed sum, Poisson summation
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
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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, Volume 32 (2020) no. 3, pp. 719-760. doi : 10.5802/jtnb.1141. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1141/

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