Far Eastern Mathematical Journal

To content of the issue


On certain Littlewood-like and Schmidt-like problems in inhomogeneous Diophantine approximations


N. G. Moshchevitin

2012, issue 2, P. 237–254


Abstract
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].

Keywords:
Diophantine approximation, Littlewood conjecture, Peres – Schlag's method, badly approximable numbers

Download the article (PDF-file)

References

[1] D. Badziahin, A. Pollington, S. Velani, “On a problem in simultaneous Diophantine approximation: Schmidt's conjecture”, Annals of Mathematics, 174 (2011), 1837–1883.
[2] D. Badziahin, On multiplicatively badly approximable numbers, 2011, arXiv: 1101.1855.
[3] Y. Bugeaud, “Multiplicative Diophantine approximation”, Dynamical systems and Diophantine Approximation, Socie?te? mathe?matique de France, Se?minaires et Congre?s 19, 2009, 107–127.
[4] Y. Bugeaud, N. Moshchevitin, “Badly approximable numbers and Littlewood-type problems”, Mathematical Proceedings of the Cambridge Philosophical Society, 150:02 (2011), 215–226.
[5] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts Math. Math. Phys., v. 45, Cambridge Univ. Press, New York, 1957.
[6] H. J. Godwin, “On the theorem of Khintchine”, Proc. London Math. Soc, 3, 1:1 (1953), 211–221.
[7] A. Khintchine, “U?ber eine Klasse linearer diophantischer Approximationen”, Rendiconti Circ. Math. Palermo, 50:2 (1926), 170–195.
[8] E. Lindenstrauss, U. Shapira, Homogeneous orbit closures and applications, 2011, arXiv: 1101.3945.
[9] N. Moshchevitin, “Khintchine's Diophantine singular systems and their applications”, Russian Math. Surveys, 65:3 (2010), 433–511.
[10] N. Moshchevitin, “On simultaneously badly approximable numbers”, Bull. London Math. Soc., 42:1 (2010), 149–154.
[11] N. Moshchevitin, “A note on badly approximable affine forms and winning sets”, Moscow Mathematical Journal, 11:1 (2011), 129–137.
[12] N. G. Moshchevitin, Schmidt's conjecture and Badziahin – Pollington – Velani's theorem, 2010, arXiv: 1004.4269.
[13] Y. Peres, W. Schlag, “Two Erdo?s problems on lacunary sequences: chromatic numbers and Diophantine approximations”, Bull. London Math. Soc., 42:2 (2010), 295–300.
[14] A. M. Rockett, P. Szu?sz, Continued Fractions, World Scientific Publishing Co., 1992.
[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, Birkha?user, 1983, 271–289.
[17] U. Shapira, “A solution to a problem of Cassels and Diophantine properties of cubic numbers”, Annals of Mathematics, 173:1 (2011), 543–557.
[18] U. Shapira, Grids with dense values, 2011, arXiv: 1101.3941.
[19] J. Tseng, “Badly approhimable affine forms and Schmidt games”, J. Number Theory, 129:12 (2009), 3020–3025.

To content of the issue