We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular matrices are both equal to , thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.
Nous établissons un nouveau lien entre la théorie métrique de l'approximation diophantienne et la géométrie paramétrique des nombres, en démontrant un principe variationnel permettant le calcul des dimensions de Hausdorff et d'entassement de nombreux ensembles d'intérêt en approximation diophantienne. Comme cas particulier, nous démontrons que les dimensions de Hausdorff et d'entassement de l'ensemble des matrices singulières de dimensions sont toutes deux égales à , démontrant ainsi une conjecture de Kadyrov, Kleinbock, Lindenstrauss et Margulis, et répondant par là même à une question soulevée par Bugeaud, Cheung et Chevallier. D'autres exemples d'application incluent le calcul des dimensions des ensembles de points satisfaisant des conjectures énoncées par Starkov et Schmidt.
Accepted:
Published online:
Tushar Das 1; Lior Fishman 2; David Simmons 3; Mariusz Urbański 2
@article{CRMATH_2017__355_8_835_0, author = {Tushar Das and Lior Fishman and David Simmons and Mariusz Urba\'nski}, title = {A variational principle in the parametric geometry of numbers, with applications to metric {Diophantine} approximation}, journal = {Comptes Rendus. Math\'ematique}, pages = {835--846}, publisher = {Elsevier}, volume = {355}, number = {8}, year = {2017}, doi = {10.1016/j.crma.2017.07.007}, language = {en}, }
TY - JOUR AU - Tushar Das AU - Lior Fishman AU - David Simmons AU - Mariusz Urbański TI - A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation JO - Comptes Rendus. Mathématique PY - 2017 SP - 835 EP - 846 VL - 355 IS - 8 PB - Elsevier DO - 10.1016/j.crma.2017.07.007 LA - en ID - CRMATH_2017__355_8_835_0 ER -
%0 Journal Article %A Tushar Das %A Lior Fishman %A David Simmons %A Mariusz Urbański %T A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation %J Comptes Rendus. Mathématique %D 2017 %P 835-846 %V 355 %N 8 %I Elsevier %R 10.1016/j.crma.2017.07.007 %G en %F CRMATH_2017__355_8_835_0
Tushar Das; Lior Fishman; David Simmons; Mariusz Urbański. A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 835-846. doi : 10.1016/j.crma.2017.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.07.007/
[1] Hausdorff dimension and uniform exponents in dimension two, 2016 | arXiv
[2] An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45, Cambridge University Press, New York, 1957
[3] Hausdorff dimension of the set of singular pairs, Ann. of Math. (2), Volume 173 (2011) no. 1, pp. 127-167
[4] Hausdorff dimension of singular vectors, Duke Math. J., Volume 165 (2016) no. 12, pp. 2273-2329 (MR 3544282)
[5] Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., Volume 359 (1985), pp. 55-89
[6] Dirichlet's theorem on Diophantine approximation. II, Acta Arith., Volume 16 (1969/1970), pp. 413-424 (MR 0279040)
[7] Zum Khintchineschen “Übertragungssatz”, Trav. Inst. Math. Tbilissi, Volume 3 (1938), pp. 193-212 (in German)
[8] Singular systems of linear forms and non-escape of mass in the space of lattices, 2014 | arXiv
[9] On a conjecture of Schmidt for the parametric geometry of numbers, Mosc. J. Comb. Number Theory, Volume 6 (2016) no. 2–3, pp. 166-176
[10] Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo, Volume 50 (1926), pp. 170-195 (in German)
[11] Über singuläre Zahlensysteme, Compos. Math., Volume 4 (1937), pp. 424-431 (MR 1556985)
[12] Regular systems of linear equations and a general problem of Čebyšev, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 12 (1948), pp. 249-258 (MR 0025513)
[13] Khintchine's singular Diophantine systems and their applications, Russ. Math. Surv., Volume 65 (2010) no. 3, pp. 433-511
[14] Proof of W.M. Schmidt's conjecture concerning successive minima of a lattice, J. Lond. Math. Soc. (2), Volume 86 (2012) no. 1, pp. 129-151 (MR 2959298)
[15] On Schmidt and Summerer parametric geometry of numbers, Ann. of Math. (2), Volume 182 (2015) no. 2, pp. 739-786 (MR 3418530)
[16] Spectrum of the exponents of best rational approximation, Math. Z., Volume 283 (2016) no. 1–2, pp. 143-155 (MR 3489062)
[17] Diophantine approximation and special Liouville numbers, Commun. Math., Volume 21 (2013) no. 1, pp. 39-76 (MR 3067121)
[18] On badly approximable numbers and certain games, Trans. Amer. Math. Soc., Volume 123 (1966), pp. 27-50
[19] Open problems in Diophantine approximation, Luminy, 1982 (Prog. Math.), Volume vol. 31, Birkhäuser Boston, Boston, MA (1983), pp. 271-287 (MR 702204)
[20] Parametric geometry of numbers and applications, Acta Arith., Volume 140 (2009) no. 1, pp. 67-91 (MR 2557854)
[21] Diophantine approximation and parametric geometry of numbers, Monatshefte Math., Volume 169 (2013) no. 1, pp. 51-104 (MR 3016519)
[22] Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, vol. 190, American Mathematical Society, Providence, RI, 2000 (translated from the 1999 Russian original by the author. MR 1746847)
[23] Computable Analysis. An Introduction, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag, Berlin, 2000 (MR 1795407)
Cited by Sources:
Comments - Policy