Liminf Sets in Simultaneous Diophantine Approximation

Faustin Adiceam

doi:10.5802/jtnb.949

Faustin Adiceam ¹

¹ Department of Mathematics, University of York YO10 5DD Heslington, United Kingdom

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 461-483.

Abstract
Résumé

Let $𝒬$ be an infinite set of positive integers. Denote by $W_{τ, n}^{*} (𝒬)$ the set of $n$ –tuples of real numbers simultaneously $τ$ –well approximable by infinitely many rationals with denominators in $𝒬$ but by only finitely many rationals with denominators in the complement of $𝒬$ . The Hausdorff dimension of the liminf set $W_{τ, n}^{*} (𝒬)$ is determined when $n \geq 1$ and $τ > 2 + 1 / n$ . A $p$ –adic analogue of the problem is also studied.

Soit $𝒬$ un ensemble infini d’entiers naturels non nuls. Soit $W_{τ, n}^{*} (𝒬)$ l’ensemble des $n$ –uplets de réels simultanément approchables à l’ordre $τ$ par une infinité de rationnels dont les dénominateurs sont dans $𝒬$ mais seulement par un nombre fini de rationnels dont les dénominateurs sont dans le complémentaire de $𝒬$ . Nous déterminons la dimension de Hausdorff de l’ensemble liminf $W_{τ, n}^{*} (𝒬)$ lorsque $n \geq 1$ et $τ > 2 + 1 / n$ . Un analogue $p$ –adique du problème considéré est également étudié.

Received: 2014-04-14
Accepted: 2014-09-04
Published online: 2017-01-02

MR Zbl

DOI: 10.5802/jtnb.949

Classification: 11J83, 11K60
Keywords: Diophantine approximation, liminf sets, Hausdorff dimension.

Author's affiliations:

Faustin Adiceam ¹

¹ Department of Mathematics, University of York YO10 5DD Heslington, United Kingdom

@article{JTNB_2016__28_2_461_0,
     author = {Faustin Adiceam},
     title = {Liminf {Sets} in {Simultaneous} {Diophantine} {Approximation}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {461--483},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
     number = {2},
     year = {2016},
     doi = {10.5802/jtnb.949},
     zbl = {1415.11102},
     mrnumber = {3509720},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.949/}
}

TY  - JOUR
AU  - Faustin Adiceam
TI  - Liminf Sets in Simultaneous Diophantine Approximation
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2016
SP  - 461
EP  - 483
VL  - 28
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.949/
DO  - 10.5802/jtnb.949
LA  - en
ID  - JTNB_2016__28_2_461_0
ER  -

%0 Journal Article
%A Faustin Adiceam
%T Liminf Sets in Simultaneous Diophantine Approximation
%J Journal de théorie des nombres de Bordeaux
%D 2016
%P 461-483
%V 28
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.949/
%R 10.5802/jtnb.949
%G en
%F JTNB_2016__28_2_461_0

Faustin Adiceam. Liminf Sets in Simultaneous Diophantine Approximation. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 2, pp. 461-483. doi : 10.5802/jtnb.949. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.949/

References
Cited by

[1] A. G. Abercrombie, « The Hausdorff dimension of some exceptional sets of $p$ -adic integer matrices », J. Number Theory 53 (1995), no. 2, p. 311-341. | DOI | MR | Zbl

[2] F. Adiceam, « A note on the Hausdorff dimension of some liminf sets appearing in simultaneous Diophantine approximation », Mathematika 59 (2013), no. 1, p. 56-64. | DOI | MR | Zbl

[3] V. Beresnevich, D. Dickinson & S. Velani, « Measure theoretic laws for lim sup sets », Mem. Amer. Math. Soc. 179 (2006), no. 846, p. x+91. | DOI | MR | Zbl

[4] V. Beresnevich, G. Harman, A. Haynes & S. Velani, « The Duffin-Schaeffer conjecture with extra divergence II », Math. Z. 275 (2013), no. 1-2, p. 127-133. | DOI | MR | Zbl

[5] A. S. Besicovitch, « Sets of Fractional Dimensions (IV): On Rational Approximation to Real Numbers », J. London Math. Soc. S1-9 (1934), no. 2, p. 126. | DOI | MR | Zbl

[6] I. Borosh & A. S. Fraenkel, « A generalization of Jarník’s theorem on Diophantine approximations », Nederl. Akad. Wetensch. Proc. Ser. A Indag. Math. 75 (1972), p. 193-201. | DOI | Zbl

[7] Y. Bugeaud & A. Durand, « Metric Diophantine approximation on the middle–third Cantor set », Accepted in J. Eur. Math. Soc. | DOI | MR | Zbl

[8] R. J. Duffin & A. C. Schaeffer, « Khintchine’s problem in metric Diophantine approximation », Duke Math. J. 8 (1941), p. 243-255. | DOI | MR | Zbl

[9] K. Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990, Mathematical foundations and applications, xxii+288 pages. | Zbl

[10] G. H. Hardy & M. Riesz, The general theory of Dirichlet’s series, Cambridge University Press, 1915. | Zbl

[11] V. Jarník, « Über die simultanen diophantischen Approximationen », Math. Z. 33 (1931), no. 1, p. 505-543. | DOI

Cited by Sources: