Abstract
Dirichlet’s Theorem on simultaneous approximation involves a parameter Q. It can be derived from Minkowski’s geometry of numbers involving a symmetric convex body depending on Q. Therefore it is natural to study parametric Geometry of Numbers. In turn, this will shed new light on diophantine approximation.
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References
Bugeaud Y., Laurent M.: On exponents of homogeneous and inhomogeneous Diophantine approximation. Mosc. Math. J. 5(4), 747–766 (2005)
Bugeaud, Y., Laurent, M.: Exponents of Diophantine Approximation. Diophantine Geometry. CRM Series 4, pp. 101–121. Ed. Normale, Pisa (2007)
Bugeaud Y., Laurent M.: On transfer inequalities in diophantine approximation II. Math. Z. 265(2), 249–262 (2010)
Jarník V.: Zum Khintchineschen Übertragungssatz. Trav. Inst. Math. Tbilissi 3, 193–212 (1938)
Jarník V.: On the theory of homogeneous linear diophantine approximations. Czech Math. J. 4(79), 330–353 (1954)
Khintchine A.Y.: Zur metrischen Theorie der Diophantischen Approximationen. Math. Z. 24, 706–714 (1926)
Khintchine, A.Y.: Über eine Klasse linearer Diophantischer Approximationen. Rend. Circ. Math. Palermo 50, 170–195
Laurent M.: Exponents of Diophantine approximation in dimension two. Can. J. Math. 61(1), 165–189 (2009)
Mahler K.: On compound convex bodies I, II. Proc. Lond. Math. Soc. 5(3), 358–384 (1955)
Minkowski H.: Geometrie der Zahlen. Teubner Verlag, Leipzig (1910)
Moshchevitin N.G.: Khintchines singular diophantine systems and their applications. Russian Math. Surveys 65(3), 433–511 (2010)
Moshchevitin, N.G.: Proof of W.M. Schmidt’s conjecture concerning successive minima of a lattice. Preprint. arXiv:0804.0120
Moshchevitin, N.G.: Exponents for three-dimensional simultaneous Diophantine approximations. Preprint. arXiv:1009.0987
Roy D.: On two exponents of approximation related to a number and its square. Can. J. Math. 59(1), 211–224 (2007)
Schmidt W.M., Summerer L.: Parametric geometry of numbers and applications. Acta Arithmetica 140(1), 67–91 (2009)
Schmidt, W.M.: Open problems in Diophantine approximation. Diophantine Approximations and Transcendental Numbers (Luminy, 1982). Progr. Math., vol. 31, pp. 271–287. Birkhäuser, Boston (1983)
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Communicated by J. Schoißengeier.
L. Summerer was supported by FWF grant P22794-N13.
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Schmidt, W.M., Summerer, L. Diophantine approximation and parametric geometry of numbers. Monatsh Math 169, 51–104 (2013). https://doi.org/10.1007/s00605-012-0391-z
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DOI: https://doi.org/10.1007/s00605-012-0391-z