Approximation diophantienne sur les quadriques
Annales Henri Lebesgue, Volume 5 (2022), pp. 1009-1034.

Metadata

Keywordsespaces de réseaux, points rationnels, groupes orthogonaux, théorème du sous-espace

Abstract

Nous étudions l’exposant diophantien d’un point x d’une hypersurface quadratique. Nous montrons notamment un analogue du théorème de Thue–Siegel–Roth, c’est-à-dire une formule pour l’exposant diophantien d’un point algébrique, et un analogue du résultat de Kleinbock et Margulis sur l’extrémalité des sous-variétés non dégénérées de l’espace affine.


References

[BCR98] Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 36, Springer, 1998 (translated from the 1987 French original, Revised by the authors) | DOI | MR | Zbl

[BdS21] Breuillard, Emmanuel; de Saxcé, Nicolas A subspace theorem for manifolds (2021) (à paraître dans JEMS, https://arxiv.org/abs/2101.04055)

[Dan85] Dani, Shrikrishna G. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89 | DOI | MR | Zbl

[Dru05] Druţu, Cornelia Diophantine approximation on rational quadrics, Math. Ann., Volume 333 (2005) no. 2, pp. 405-470 | DOI | MR | Zbl

[dS] de Saxcé, Nicolas Non divergence via the successive minima (à paraître dans Groups, Geometry and Dynamics)

[FKMS22] Fishman, Lior; Kleinbock, Dmitry; Merrill, Keith; Simmons, David Intrinsic Diophantine approximation on quadric hypersurfaces, J. Eur. Math. Soc., Volume 24 (2022) no. 3, pp. 1045-1101 | DOI | MR | Zbl

[Kle08] Kleinbock, Dmitry An extension of quantitative nondivergence and applications to Diophantine exponents, Trans. Am. Math. Soc., Volume 360 (2008) no. 12, pp. 6497-6523 | DOI | MR | Zbl

[Kle10] Kleinbock, Dmitry An “almost all versus no” dichotomy in homogeneous dynamics and Diophantine approximation, Geom. Dedicata, Volume 149 (2010), pp. 205-218 | DOI | MR | Zbl

[KLW04] Kleinbock, Dmitry; Lindenstrauss, Elon; Weiss, Barak On fractal measures and Diophantine approximation, Sel. Math., New Ser., Volume 10 (2004) no. 4, pp. 479-523 | DOI | MR | Zbl

[KM98] Kleinbock, Dmitry; Margulis, Grigoriĭ A. Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math., Volume 148 (1998) no. 1, pp. 339-360 | DOI | MR | Zbl

[KM15] Kleinbock, Dmitry; Merrill, Keith Rational approximation on spheres, Isr. J. Math., Volume 209 (2015) no. 1, pp. 293-322 | DOI | MR | Zbl

[Mar71] Margulis, Grigoriĭ A. The action of unipotent groups in a lattice space, Mat. Sb., N. Ser., Volume 86(128) (1971), pp. 552-556 | MR

[Rot60] Roth, Klaus F., Proc. Internat. Congress Math. 1958 (1960), pp. 203-210 | Zbl

[Sch80] Schmidt, Wolfgang M. Diophantine approximation, Lecture Notes in Mathematics, 785, Springer, 1980 | MR | Zbl