On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations
Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 237-254 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We give several results related to inhomogeneous approximations to two real numbers and badly approximable numbers. Our results are related to classical theorems by A. Khintchine [7] and to an original method invented by Y. Peres and W. Schlag [13].
@article{DVMG_2012_12_2_a9,
     author = {N. G. Moshchevitin},
     title = {On certain {Littlewood-like} and {Schmidt-like} problems in inhomogeneous {Diophantine} approximations},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {237--254},
     year = {2012},
     volume = {12},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a9/}
}
TY  - JOUR
AU  - N. G. Moshchevitin
TI  - On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2012
SP  - 237
EP  - 254
VL  - 12
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a9/
LA  - en
ID  - DVMG_2012_12_2_a9
ER  - 
%0 Journal Article
%A N. G. Moshchevitin
%T On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2012
%P 237-254
%V 12
%N 2
%U http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a9/
%G en
%F DVMG_2012_12_2_a9
N. G. Moshchevitin. On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations. Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 2, pp. 237-254. http://geodesic.mathdoc.fr/item/DVMG_2012_12_2_a9/

[1] D. Badziahin, A. Pollington, S. Velani, “On a problem in simultaneous Diophantine approximation: Schmidt's conjecture”, Annals of Mathematics, 174 (2011), 1837–1883 | DOI | MR | Zbl

[2] D. Badziahin, On multiplicatively badly approximable numbers, 2011, arXiv: 1101.1855 | MR

[3] Y. Bugeaud, “Multiplicative Diophantine approximation”, Dynamical systems and Diophantine Approximation, Société mathématique de France, Séminaires et Congrès 19, 2009, 107–127 | MR

[4] Y. Bugeaud, N. Moshchevitin, “Badly approximable numbers and Littlewood-type problems”, Mathematical Proceedings of the Cambridge Philosophical Society, 150:02 (2011), 215–226 | DOI | MR | Zbl

[5] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts Math. Math. Phys., v. 45, Cambridge Univ. Press, New York, 1957 | MR | Zbl

[6] H. J. Godwin, “On the theorem of Khintchine”, Proc. London Math. Soc, 3, 1:1 (1953), 211–221 | DOI | MR | Zbl

[7] A. Khintchine, “Über eine Klasse linearer diophantischer Approximationen”, Rendiconti Circ. Math. Palermo, 50:2 (1926), 170–195 | DOI | Zbl

[8] E. Lindenstrauss, U. Shapira, Homogeneous orbit closures and applications, 2011, arXiv: 1101.3945 | MR

[9] N. Moshchevitin, “Khintchine's Diophantine singular systems and their applications”, Russian Math. Surveys, 65:3 (2010), 433–511 | DOI | MR | Zbl

[10] N. Moshchevitin, “On simultaneously badly approximable numbers”, Bull. London Math. Soc., 42:1 (2010), 149–154 | DOI | MR | Zbl

[11] N. Moshchevitin, “A note on badly approximable affine forms and winning sets”, Moscow Mathematical Journal, 11:1 (2011), 129–137 | MR | Zbl

[12] N. G. Moshchevitin, Schmidt's conjecture and Badziahin – Pollington – Velani's theorem, 2010, arXiv: 1004.4269

[13] Y. Peres, W. Schlag, “Two Erdös problems on lacunary sequences: chromatic numbers and Diophantine approximations”, Bull. London Math. Soc., 42:2 (2010), 295–300 | DOI | MR | Zbl

[14] A. M. Rockett, P. Szüsz, Continued Fractions, World Scientific Publishing Co., 1992 | MR | Zbl

[15] W. M. Schmidt, Diophantine Approximations, Lect. Not. Math, v. 785, 1980

[16] W. M. Schmidt, “Open problems in Diophantine approximations”, Approximations Diophantiennes et nombres transcendants' Luminy, 1982, Progress in Mathematics, Birkhäuser, 1983, 271–289 | MR

[17] U. Shapira, “A solution to a problem of Cassels and Diophantine properties of cubic numbers”, Annals of Mathematics, 173:1 (2011), 543–557 | DOI | MR | Zbl

[18] U. Shapira, Grids with dense values, 2011, arXiv: 1101.3941 | MR

[19] J. Tseng, “Badly approhimable affine forms and Schmidt games”, J. Number Theory, 129:12 (2009), 3020–3025 | DOI | MR | Zbl