Abstract
We prove that almost any pair of real numbers $\alpha,\beta$, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: $$\begin{align}\label{C1abst}\tag{C1} \forall \gamma,\delta\in\mathbb{R},\quad \liminf_{n\to\infty} \leftn\right\langle n\alpha\gamma \rangle\langle n\beta\delta\rangle=0, \end{align}$$ where $\langle\cdot\rangle$ denotes the distance from the nearest integer. The existence of even a single pair that satisfies statement (C1), solves a problem of Cassels from the 50’s. We then prove that if $1,\alpha,\beta$ span a totally real cubic number field, then $\alpha,\beta$, satisfy (C1). This generalizes a result of Cassels and SwinnertonDyer, which says that such pairs satisfy Littlewood’s conjecture. It is further shown that if $\alpha,\beta$ are any two real numbers, such that $1,\alpha,\beta$, are linearly dependent over $\mathbb{Q}$, they cannot satisfy (C1). The results are then applied to give examples of irregular orbit closures of the diagonal group of a new type. The results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces.

[B] D. Berend, "Multiinvariant sets on tori," Trans. Amer. Math. Soc., vol. 280, iss. 2, pp. 509532, 1983.
@article {B, MRKEY = {716835},
AUTHOR = {Berend, Daniel},
TITLE = {Multiinvariant sets on tori},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {280},
YEAR = {1983},
NUMBER = {2},
PAGES = {509532},
ISSN = {00029947},
CODEN = {TAMTAM},
MRCLASS = {11K06 (11K55 28D10 54A15)},
MRNUMBER = {0716835},
MRREVIEWER = {G{é}rard Rauzy},
DOI = {10.2307/1999631},
ZBLNUMBER = {0532.10028},
} 
[Ba] E. S. Barnes, "The inhomogeneous minima of indefinite quadratic forms," J. Austral. Math. Soc., vol. 2, pp. 910, 1961/1962.
@article {Ba, MRKEY = {0124296},
AUTHOR = {Barnes, E. S.},
TITLE = {The inhomogeneous minima of indefinite quadratic forms},
JOURNAL = {J. Austral. Math. Soc.},
FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics},
VOLUME = {2},
YEAR = {1961/1962},
PAGES = {910},
ISSN = {02636115},
MRCLASS = {10.25},
MRNUMBER = {0124296},
MRREVIEWER = {R. P. Bambah},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0097.26202},
} 
[Bac] P. Bachmann, Die Arithmetik der Quadratischen FormenLeipzig and Berlin: Teubner, 1923.
@misc{Bac,
author={Bachmann, P.},
TITLE={Die Arithmetik der Quadratischen Formen},
PUBLISHER={Teubner},
ADDRESS={Leipzig and Berlin},
YEAR={1923},
NOTE={especially Kap. 12 (Die zerlegbaren formen)},
} 
[Bu] Y. Bugeaud, "Multiplicative Diophantine approximation," in Dynamical Systems and Diophantine Approximation, Paris: Soc. Math. France.
@incollection{Bu,
author={Bugeaud, Y.},
TITLE={Multiplicative Diophantine approximation},
BOOKTITLE={Dynamical {{S}}ystems and {{D}}iophantine {{A}}pproximation},
NOTE={proceedings of conference held at the Institute Henri Poincaré, Sem. et Congress, to appear},
PUBLISHER={Soc. Math. France},
ADDRESS={Paris},
} 
[BugeaudEtAlSrinking] Y. Bugeaud, S. Harrap, S. Kristensen, and S. Velani, "On shrinking targets for $\Bbb Z^m$ actions on tori," Mathematika, vol. 56, pp. 193202, 2010.
@article{BugeaudEtAlSrinking,
author={Bugeaud, Y. and Harrap, S. and Kristensen, S. and Velani, S.},
TITLE={On shrinking targets for {$\Bbb Z^m$} actions on tori},
JOURNAL={Mathematika},
VOLUME={56},
YEAR={2010},
PAGES={193202},
MRNUMBER={2678024},
} 
[Ca] J. W. S. Cassels, An Introduction to the Geometry of Numbers, New York: SpringerVerlag, 1997.
@book {Ca, MRKEY = {1434478},
AUTHOR = {Cassels, J. W. S.},
TITLE = {An {{I}}ntroduction to the {{G}}eometry of {{N}}umbers},
SERIES = {Classics Math.},
NOTE = {corrected reprint of the 1971 edition},
PUBLISHER = {SpringerVerlag},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {viii+344},
ISBN = {3540617884},
MRCLASS = {11Hxx},
MRNUMBER = {1434478},
ZBLNUMBER = {0866.11041},
} 
[Ca2] J. W. S. Cassels, "The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms," Proc. Cambridge Philos. Soc., vol. 48, pp. 7286, 1952.
@article {Ca2, MRKEY = {0047709},
AUTHOR = {Cassels, J. W. S.},
TITLE = {The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms},
JOURNAL = {Proc. Cambridge Philos. Soc.},
VOLUME = {48},
YEAR = {1952},
PAGES = {7286},
MRCLASS = {10.0X},
MRNUMBER = {0047709},
ZBLNUMBER={0046.04601},
MRREVIEWER = {R. Hull},
} 
[Cerri] J. P. Cerri, "Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than $1$," J. Reine Angew. Math, vol. 592, pp. 4962, 2006.
@article{Cerri,
author={Cerri, J.P.},
TITLE={Inhomogeneous and {E}uclidean spectra of number fields with unit rank strictly greater than $1$},
JOURNAL={J. Reine Angew. Math},
VOLUME={592},
YEAR={2006},
PAGES={4962},
MRNUMBER={2222729},
} 
[CaSD] J. W. S. Cassels and H. P. F. SwinnertonDyer, "On the product of three homogeneous linear forms and the indefinite ternary quadratic forms," Philos. Trans. Roy. Soc. London. Ser. A., vol. 248, pp. 7396, 1955.
@article {CaSD, MRKEY = {0070653},
AUTHOR = {Cassels, J. W. S. and SwinnertonDyer, H. P. F.},
TITLE = {On the product of three homogeneous linear forms and the indefinite ternary quadratic forms},
JOURNAL = {Philos. Trans. Roy. Soc. London. Ser. A.},
FJOURNAL = {Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences},
VOLUME = {248},
YEAR = {1955},
PAGES = {7396},
ISSN = {00804614},
MRCLASS = {10.0X},
MRNUMBER = {0070653},
ZBLNUMBER={0065.27905},
MRREVIEWER = {J. F. Koksma},
DOI = {10.1098/rsta.1955.0010},
} 
[D] H. Davenport, "Indefinite binary quadratic forms, and Euclid’s algorithm in real quadratic fields," Proc. London Math. Soc., vol. 53, pp. 6582, 1951.
@article {D, MRKEY = {0041883},
AUTHOR = {Davenport, H.},
TITLE = {Indefinite binary quadratic forms, and {E}uclid's algorithm in real quadratic fields},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Second Series},
VOLUME = {53},
YEAR = {1951},
PAGES = {6582},
ISSN = {00246115},
MRCLASS = {10.0X},
MRNUMBER = {0041883},
ZBLNUMBER={0045.01402 },
MRREVIEWER = {R. Hull},
DOI = {10.1112/plms/s253.1.65},
} 
[F] H. Furstenberg, "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation," Math. Systems Theory, vol. 1, pp. 149, 1967.
@article {F, MRKEY = {0213508},
AUTHOR = {Furstenberg, Harry},
TITLE = {Disjointness in ergodic theory, minimal sets, and a problem in {D}iophantine approximation},
JOURNAL = {Math. Systems Theory},
FJOURNAL = {Mathematical Systems Theory. An International Journal on Mathematical Computing Theory},
VOLUME = {1},
YEAR = {1967},
PAGES = {149},
ISSN = {00255661},
MRCLASS = {28.70 (10.00)},
MRNUMBER = {021350},
ZBLNUMBER={0146.28502},
MRREVIEWER = {W. Parry},
DOI = {10.1007/BF01692494},
} 
[KleinbockBadlyAppSysJNT] D. Kleinbock, "Badly approximable systems of affine forms," J. Number Theory, vol. 79, pp. 83102, 1999.
@article{KleinbockBadlyAppSysJNT,
author={Kleinbock, D.},
TITLE={Badly approximable systems of affine forms},
JOURNAL={J. Number Theory},
VOLUME={79},
YEAR={1999},
PAGES={83102},
MRNUMBER={1724255},
} 
[LW] E. Lindenstrauss and B. Weiss, "On sets invariant under the action of the diagonal group," Ergodic Theory Dynam. Systems, vol. 21, iss. 5, pp. 14811500, 2001.
@article {LW, MRKEY = {1855843},
AUTHOR = {Lindenstrauss, Elon and Weiss, Barak},
TITLE = {On sets invariant under the action of the diagonal group},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {21},
YEAR = {2001},
NUMBER = {5},
PAGES = {14811500},
ISSN = {01433857},
MRCLASS = {22E40 (22D40 37A15)},
MRNUMBER = {1855843},
ZBLNUMBER={1073.37006},
MRREVIEWER = {S. G. Dani},
DOI = {10.1017/S0143385701001717},
} 
[Ma] G. Margulis, "Problems and conjectures in rigidity theory," in Mathematics: Frontiers and Perspectives, Providence, RI: Amer. Math. Soc., 2000, pp. 161174.
@incollection {Ma, MRKEY = {1754775},
AUTHOR = {Margulis, Gregory},
TITLE = {Problems and conjectures in rigidity theory},
BOOKTITLE = {Mathematics: {{F}}rontiers and {{P}}erspectives},
PAGES = {161174},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2000},
MRCLASS = {22E40 (37C85 37D20 53C24)},
MRNUMBER = {1754775},
ZBLNUMBER={0952.22005},
MRREVIEWER = {A. I. Danilenko},
} 
[Mau] F. Maucourant, "A nonhomogeneous orbit closure of a diagonal subgroup," Ann. of Math., vol. 171, iss. 1, pp. 557570, 2010.
@article {Mau, MRKEY = {2630049},
AUTHOR = {Maucourant, Fran{ç}ois},
TITLE = {A nonhomogeneous orbit closure of a diagonal subgroup},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {171},
YEAR = {2010},
NUMBER = {1},
PAGES = {557570},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {22Exx},
MRNUMBER = {2630049},
ZBLNUMBER={1192.22006},
DOI = {10.4007/annals.2010.171.557},
} 
[R] M. Ratner, "Raghunathan’s topological conjecture and distributions of unipotent flows," Duke Math. J., vol. 63, iss. 1, pp. 235280, 1991.
@article {R, MRKEY = {1106945},
AUTHOR = {Ratner, Marina},
TITLE = {Raghunathan's topological conjecture and distributions of unipotent flows},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {63},
YEAR = {1991},
NUMBER = {1},
PAGES = {235280},
ISSN = {00127094},
CODEN = {DUMJAO},
MRCLASS = {22E40 (22D40 28D10)},
MRNUMBER = {1106945 },
MRREVIEWER = {Gopal Prasad},
DOI = {10.1215/S0012709491063118},
ZBLNUMBER = {0733.22007},
} 
[Mc] C. T. McMullen, "Minkowski’s conjecture, wellrounded lattices and topological dimension," J. Amer. Math. Soc., vol. 18, iss. 3, pp. 711734, 2005.
@article {Mc, MRKEY = {2138142},
AUTHOR = {McMullen, Curtis T.},
TITLE = {Minkowski's conjecture, wellrounded lattices and topological dimension},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {18},
YEAR = {2005},
NUMBER = {3},
PAGES = {711734},
ISSN = {08940347},
MRCLASS = {11H31 (11E57 11J83 55M10)},
MRNUMBER = {2138142},
MRREVIEWER = {Stefan K{ü}hnlein},
DOI = {10.1090/S0894034705004832},
ZBLNUMBER = {1132.11034},
} 
[Shah91] N. A. Shah, "Uniformly distributed orbits of certain flows on homogeneous spaces," Math. Ann., vol. 289, pp. 315334, 1991.
@article{Shah91,
author={Shah, N. A.},
TITLE={Uniformly distributed orbits of certain flows on homogeneous spaces},
JOURNAL={Math. Ann.},
VOLUME={289},
YEAR={1991},
PAGES={315334},
MRNUMBER={1092178},
} 
[Sh] U. Shapira, "On a generalization of Littlewood’s conjecture," J. Mod. Dyn., vol. 3, iss. 3, pp. 457477, 2009.
@article {Sh, MRKEY = {2538476},
AUTHOR = {Shapira, Uri},
TITLE = {On a generalization of {L}ittlewood's conjecture},
JOURNAL = {J. Mod. Dyn.},
FJOURNAL = {Journal of Modern Dynamics},
VOLUME = {3},
YEAR = {2009},
NUMBER = {3},
PAGES = {457477},
ISSN = {19305311},
MRCLASS = {37A17 (11J20 37A45)},
MRNUMBER = {2538476},
MRREVIEWER = {Thomas Ward},
DOI = {10.3934/jmd.2009.3.457},
ZBLNUMBER = {1185.37008},
} 
[Ts] M. Einsiedler and J. Tseng, Badly approximable systems of affine forms.
@misc{Ts,
author={Einsiedler, M. and Tseng, J.},
TITLE={Badly approximable systems of affine forms},
NOTE={preprint},
} 
[TW] G. Tomanov and B. Weiss, "Closed orbits for actions of maximal tori on homogeneous spaces," Duke Math. J., vol. 119, pp. 367392, 2003.
@article{TW,
author={Tomanov, G. and Weiss, B.},
TITLE={Closed orbits for actions of maximal tori on homogeneous spaces},
JOURNAL={Duke Math. J.},
VOLUME={119},
YEAR={2003},
PAGES={367392},
MRNUMBER={1997950},
} 
[TsengBadlyAppSysJNT] J. Tseng, "Badly approximable affine forms and Schmidt games," J. Number Theory, vol. 129, pp. 30203025, 2009.
@article{TsengBadlyAppSysJNT,
author={Tseng, J.},
TITLE={Badly approximable affine forms and {S}chmidt games},
JOURNAL={J. Number Theory},
VOLUME={129},
YEAR={2009},
PAGES={30203025},
MRNUMBER={2560849},
}