The Minkowski chain and Diophantine approximation
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 503-523.

La chaîne de Hurwitz donne une suite de paires d’approximations de Farey d’un nombre réel irrationnel. Minkowski a donné un critère d’algébraicité d’un nombre en utilisant une certaine généralisation de la chaîne de Hurwitz. Nous appliquons cette généralisation (la chaîne de Minkowski) pour donner des critères pour qu’une forme linéaire réelle soit mal approchable ou singulière. Les preuves reposent sur des propriétés des minima successifs et des bases réduites de réseaux.

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski’s generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. The proofs rely on properties of successive minima and reduced bases of lattices.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1132
Classification : 11J13, 11J70
Mots clés : Minkowski chain, Diophantine approximation
Andersen, Nickolas 1 ; Duke, William 2

1 Brigham Young University Department Of Mathematics Provo, UT 84602, USA
2 UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA
@article{JTNB_2020__32_2_503_0,
     author = {Andersen, Nickolas and Duke, William},
     title = {The {Minkowski} chain and {Diophantine} approximation},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {503--523},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {2},
     year = {2020},
     doi = {10.5802/jtnb.1132},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1132/}
}
TY  - JOUR
AU  - Andersen, Nickolas
AU  - Duke, William
TI  - The Minkowski chain and Diophantine approximation
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2020
SP  - 503
EP  - 523
VL  - 32
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1132/
DO  - 10.5802/jtnb.1132
LA  - en
ID  - JTNB_2020__32_2_503_0
ER  - 
%0 Journal Article
%A Andersen, Nickolas
%A Duke, William
%T The Minkowski chain and Diophantine approximation
%J Journal de théorie des nombres de Bordeaux
%D 2020
%P 503-523
%V 32
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1132/
%R 10.5802/jtnb.1132
%G en
%F JTNB_2020__32_2_503_0
Andersen, Nickolas; Duke, William. The Minkowski chain and Diophantine approximation. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 503-523. doi : 10.5802/jtnb.1132. http://www.numdam.org/articles/10.5802/jtnb.1132/

[1] Bugeaud, Yann On simultaneous rational approximation to a real number and its integral powers, Ann. Inst. Fourier, Volume 60 (2010) no. 6, pp. 2165-2182 | DOI | Numdam | MR | Zbl

[2] Cassels, J. W. S. An introduction to the geometry of numbers, Grundlehren der Mathematischen Wissenschaften, 99, Springer, 1959, viii+344 pages | Zbl

[3] Cheung, Yitwah Hausdorff dimension of the set of singular pairs, Ann. Math., Volume 173 (2011) no. 1, pp. 127-167 | DOI | MR | Zbl

[4] Cheung, Yitwah; Chevallier, Nicolas Hausdorff dimension of singular vectors, Duke Math. J., Volume 165 (2016) no. 12, pp. 2273-2329 | DOI | MR | Zbl

[5] Dani, Shrikrishna G. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89 | MR | Zbl

[6] Das, Tushar; Fishman, Lior; Simmons, David; Urbański, Mariusz A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 8, pp. 835-846 | MR | Zbl

[7] Davenport, Harold Minkowski’s inequality for the minima associated with a convex body, Q. J. Math., Oxf. Ser., Volume 10 (1939) no. 1, pp. 119-121 | DOI | Zbl

[8] Davenport, Harold; Schmidt, Wolfgang M. Dirichlet’s theorem on Diophantine approximation. II, Acta Arith., Volume 16 (1969), pp. 413-424 | DOI | MR | Zbl

[9] Davenport, Harold; Schmidt, Wolfgang M. Dirichlet’s theorem on Diophantine approximation, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press Inc., 1970, pp. 113-132 | Zbl

[10] Gruber, Peter M.; Lekkerkerker, Cornelis G. Geometry of numbers, North-Holland Mathematical Library, North-Holland, 1987, xvi+732 pages | Zbl

[11] Hancock, Harris Development of the Minkowski geometry of numbers. Vol. 1, 2, Dover Publications, 1964 | MR | Zbl

[12] Hurwitz, Adolf Über die angenäherte Darstellung der Zahlen durch rationale Brüche, Math. Ann., Volume XLIV (1894), pp. 417-436 | DOI | Zbl

[13] Jacobi, Carl G. J. Allgemeine Theorie der kettenbruchähnlichen Algorithmen, in welche jede Zahl aus drei vorhergehenden gebildet wird, J. Reine Angew. Math., Volume 69 (1891), pp. 29-64

[14] Khintchine, Aleksandr Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo, Volume 50 (1926), pp. 170-195 | DOI | Zbl

[15] Khintchine, Aleksandr Zur metrischen Theorie der diophantischen Approximationen, Math. Z., Volume 24 (1926), pp. 706-714 | DOI | MR | Zbl

[16] Lagarias, Jeffrey C. Number theory and dynamical systems, The unreasonable effectiveness of number theory (Orono, ME, 1991) (Proceedings of Symposia in Applied Mathematics), Volume 46, American Mathematical Society, 1991, pp. 35-72 | DOI | Zbl

[17] Lagarias, Jeffrey C. Geodesic multidimensional continued fractions, Proc. Lond. Math. Soc., Volume 69 (1994) no. 3, pp. 464-488 | DOI | MR | Zbl

[18] Mahler, Kurt On Minkowski’s theory of reduction of positive definite quadratic forms, Q. J. Math., Oxf. Ser., Volume 9 (1938), pp. 259-262 | DOI | Zbl

[19] Minkowski, Hermann Ein Kriterium für die algebraischen Zahlen, Gött. Nachr., Volume 1899 (1899), pp. 64-88 | Zbl

[20] Minkowski, Hermann Über periodische Approximationen algebraischer Zahlen, Acta Math. (1902), pp. 333-352 | DOI | MR | Zbl

[21] Minkowski, Hermann Diskontinuitatsbereich fur arithmetische Aquivalenz, J. Reine Angew. Math., Volume 129 (1905), pp. 220-274 | DOI | MR

[22] Minkowski, Hermann Geometrie der Zahlen, Teubner, 1910 | Zbl

[23] Philippon, Patrice A Farey tail, Notices Am. Math. Soc., Volume 59 (2012) no. 6, pp. 746-757 | DOI | MR | Zbl

[24] Schmidt, Wolfgang M. Badly approximable systems of linear forms, J. Number Theory, Volume 1 (1969), pp. 139-154 | DOI | MR | Zbl

[25] Schmidt, Wolfgang M. Diophantine approximation, Lecture Notes in Mathematics, Springer, 1980, x+299 pages | Zbl

[26] Schmidt, Wolfgang M.; Summerer, Leonhard Parametric geometry of numbers and applications, Acta Arith., Volume 140 (2009) no. 1, pp. 67-91 | DOI | MR | Zbl

[27] Siegel, Carl L. Lectures on the geometry of numbers, Springer, 1989, x+160 pages

[28] Weyl, Hermann On geometry of numbers, Proc. Lond. Math. Soc., Volume 47 (1942), pp. 268-289 | DOI | MR | Zbl

[29] Weyl, Hermann; Siegel, Carl L.; Mahler, Kurt Seminar on Geometry of Numbers, IAS, 1949

Cité par Sources :