The Minkowski chain and Diophantine approximation
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 503-523.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1132
Classification: 11J13,  11J70
Keywords: Minkowski chain, Diophantine approximation
Nickolas Andersen 1; William Duke 2

1 Brigham Young University Department Of Mathematics Provo, UT 84602, USA
2 UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555, USA
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Nickolas Andersen; William Duke. The Minkowski chain and Diophantine approximation. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 2, pp. 503-523. doi : 10.5802/jtnb.1132. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1132/

[1] Yann Bugeaud On simultaneous rational approximation to a real number and its integral powers, Ann. Inst. Fourier, Volume 60 (2010) no. 6, pp. 2165-2182 | Article | Numdam | MR: 2791654 | Zbl: 1229.11100

[2] J. W. S. Cassels An introduction to the geometry of numbers, Grundlehren der Mathematischen Wissenschaften, Volume 99, Springer, 1959, viii+344 pages | Zbl: 0086.26203

[3] Yitwah Cheung Hausdorff dimension of the set of singular pairs, Ann. Math., Volume 173 (2011) no. 1, pp. 127-167 | Article | MR: 2753601 | Zbl: 1241.11075

[4] Yitwah Cheung; Nicolas Chevallier Hausdorff dimension of singular vectors, Duke Math. J., Volume 165 (2016) no. 12, pp. 2273-2329 | Article | MR: 3544282 | Zbl: 1358.11078

[5] Shrikrishna G. Dani Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89 | MR: 794799 | Zbl: 0578.22012

[6] Tushar Das; Lior Fishman; David Simmons; Mariusz Urbański 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: 3693502 | Zbl: 1427.11072

[7] Harold Davenport Minkowski’s inequality for the minima associated with a convex body, Q. J. Math., Oxf. Ser., Volume 10 (1939) no. 1, pp. 119-121 | Article | Zbl: 65.0175.01

[8] Harold Davenport; Wolfgang M. Schmidt Dirichlet’s theorem on Diophantine approximation. II, Acta Arith., Volume 16 (1969), pp. 413-424 | Article | MR: 279040 | Zbl: 0201.05501

[9] Harold Davenport; Wolfgang M. Schmidt Dirichlet’s theorem on Diophantine approximation, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press Inc., 1970, pp. 113-132 | Zbl: 0226.10032

[10] Peter M. Gruber; Cornelis G. Lekkerkerker Geometry of numbers, North-Holland Mathematical Library, North-Holland, 1987, xvi+732 pages | Zbl: 0611.10017

[11] Harris Hancock Development of the Minkowski geometry of numbers. Vol. 1, 2, Dover Publications, 1964 | MR: 169821 | Zbl: 0123.25603

[12] Adolf Hurwitz Über die angenäherte Darstellung der Zahlen durch rationale Brüche, Math. Ann., Volume XLIV (1894), pp. 417-436 | Article | Zbl: 25.0322.04

[13] Carl G. J. Jacobi 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] Aleksandr Khintchine Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo, Volume 50 (1926), pp. 170-195 | Article | Zbl: 52.0183.01

[15] Aleksandr Khintchine Zur metrischen Theorie der diophantischen Approximationen, Math. Z., Volume 24 (1926), pp. 706-714 | Article | MR: 1544787 | Zbl: 52.0183.02

[16] Jeffrey C. Lagarias 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 | Article | Zbl: 0773.11051

[17] Jeffrey C. Lagarias Geodesic multidimensional continued fractions, Proc. Lond. Math. Soc., Volume 69 (1994) no. 3, pp. 464-488 | Article | MR: 1289860 | Zbl: 0813.11040

[18] Kurt Mahler On Minkowski’s theory of reduction of positive definite quadratic forms, Q. J. Math., Oxf. Ser., Volume 9 (1938), pp. 259-262 | Article | Zbl: 0019.39503

[19] Hermann Minkowski Ein Kriterium für die algebraischen Zahlen, Gött. Nachr., Volume 1899 (1899), pp. 64-88 | Zbl: 30.0195.02

[20] Hermann Minkowski Über periodische Approximationen algebraischer Zahlen, Acta Math. (1902), pp. 333-352 | Article | MR: 1554968 | Zbl: 33.0216.02

[21] Hermann Minkowski Diskontinuitatsbereich fur arithmetische Aquivalenz, J. Reine Angew. Math., Volume 129 (1905), pp. 220-274 | Article | MR: 1580668

[22] Hermann Minkowski Geometrie der Zahlen, Teubner, 1910 | Zbl: 41.0239.03

[23] Patrice Philippon A Farey tail, Notices Am. Math. Soc., Volume 59 (2012) no. 6, pp. 746-757 | Article | MR: 2977610 | Zbl: 1351.11016

[24] Wolfgang M. Schmidt Badly approximable systems of linear forms, J. Number Theory, Volume 1 (1969), pp. 139-154 | Article | MR: 248090 | Zbl: 0172.06401

[25] Wolfgang M. Schmidt Diophantine approximation, Lecture Notes in Mathematics, Springer, 1980, x+299 pages | Zbl: 0421.10019

[26] Wolfgang M. Schmidt; Leonhard Summerer Parametric geometry of numbers and applications, Acta Arith., Volume 140 (2009) no. 1, pp. 67-91 | Article | MR: 2557854 | Zbl: 1236.11060

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

[28] Hermann Weyl On geometry of numbers, Proc. Lond. Math. Soc., Volume 47 (1942), pp. 268-289 | Article | MR: 6212 | Zbl: 0028.34902

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

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