Geodesic
Transference inequalities for multiplicative Diophantine exponents
Informatics and Automation, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 227-239.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove inequalities for multiplicative analogues of Diophantine exponents; these inequalities are similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly approximable and establish some inequalities connecting multiplicative exponents with ordinary ones. 
@article{TRSPY_2011_275_a14,
     author = {Oleg N. German},
     title = {Transference inequalities for multiplicative {Diophantine} exponents},
     journal = {Informatics and Automation},
     pages = {227--239},
     publisher = {mathdoc},
     volume = {275},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2011_275_a14/}
}
TY  - JOUR
AU  - Oleg N. German
TI  - Transference inequalities for multiplicative Diophantine exponents
JO  - Informatics and Automation
PY  - 2011
SP  - 227
EP  - 239
VL  - 275
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2011_275_a14/
LA  - en
ID  - TRSPY_2011_275_a14
ER  - 
%0 Journal Article
%A Oleg N. German
%T Transference inequalities for multiplicative Diophantine exponents
%J Informatics and Automation
%D 2011
%P 227-239
%V 275
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2011_275_a14/
%G en
%F TRSPY_2011_275_a14
Oleg N. German. Transference inequalities for multiplicative Diophantine exponents. Informatics and Automation, Classical and modern mathematics in the wake of Boris Nikolaevich Delone, Tome 275 (2011), pp. 227-239. http://geodesic.mathdoc.fr/item/TRSPY_2011_275_a14/

[1] Schmidt W.M., Wang Y., “A note on a transference theorem of linear forms”, Sci. Sin., 22:3 (1979), 276–280 | MR | Zbl

[2] Wang Y., Yu K., “A note on some metrical theorems in Diophantine approximation”, Chin. Ann. Math., 2 (1981), 1–12 | MR | Zbl

[3] Cassels J.W.S., An introduction to Diophantine approximation, Cambridge Univ. Press, Cambridge, 1957 | MR | Zbl

[4] Cassels J.W.S., Swinnerton-Dyer H.P.F., “On the product of three homogeneous linear forms and indefinite ternary quadratic forms”, Philos. Trans. Roy. Soc. London A, 248 (1955), 73–96 | DOI | MR | Zbl

[5] Dyson F.J., “On simultaneous Diophantine approximations”, Proc. London Math. Soc. Ser. 2, 49 (1947), 409–420 | DOI | MR | Zbl

[6] Mahler K., “Ein Übertragungsprinzip für lineare Ungleichungen”, Čas. Pěst. Mat. Fys., 68 (1939), 85–92 | MR | Zbl

[7] Maler K., “Ob odnoi teoreme Daisona”, Mat. sb., 26:3 (1950), 457–462 | MR

[8] Khintchine A.Ya., “Über eine Klasse linearer diophantischer Approximationen”, Rend. Circ. Mat. Palermo, 50 (1926), 170–195 | DOI | Zbl

[9] Bugeaud Y., “Multiplicative Diophantine approximation”, Dynamical systems and Diophantine approximation, Proc. Conf. Inst. H. Poincaré, Soc. math. France, Paris, 2009, 105–125 | MR | Zbl

[10] German O.N., On Diophantine exponents and Khintchine's transference principle, E-print, 2010

[11] Vaaler J.D., “A geometric inequality with applications to linear forms”, Pac. J. Math., 83:2 (1979), 543–553 | DOI | MR | Zbl

[12] Ball K., “Volumes of sections of cubes and related problems”, Geometric aspects of functional analysis, Isr. Semin., GAFA, 1987–1988, Lect. Notes Math., 1376, Springer, Berlin, 1989, 251–260 | DOI | MR

[13] Bonnesen T., Fenchel W., Theorie der konvexen Körper, Springer, Berlin, 1934 | MR | Zbl

[14] Jarník V., “Zum Khintchineschen “Übertragungssatz””, Trav. Inst. Math. Tbilissi, 3 (1938), 193–212 | Zbl