Abstract
We prove that almost any pair of real numbers $\alpha,\beta$, satisfies the following inhomogeneous uniform version of Littlewood’s conjecture: $$\begin{align}\label{C1abst}\tag{C1} \forall \gamma,\delta\in\mathbb{R},\quad \liminf_{|n|\to\infty} \left|n\right|\langle n\alpha-\gamma \rangle\langle n\beta-\delta\rangle=0, \end{align}$$ 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 Swinnerton-Dyer, 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, "Multi-invariant sets on tori," Trans. Amer. Math. Soc., vol. 280, iss. 2, pp. 509-532, 1983.@article {B, MRKEY = {716835},
AUTHOR = {Berend, Daniel},
TITLE = {Multi-invariant sets on tori},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {280},
YEAR = {1983},
NUMBER = {2},
PAGES = {509--532},
ISSN = {0002-9947},
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. 9-10, 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 = {9--10},
ISSN = {0263-6115},
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. 193-202, 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={193--202},
MRNUMBER={2678024},
} -
[Ca] J. W. S. Cassels, An Introduction to the Geometry of Numbers, New York: Springer-Verlag, 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 = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1997},
PAGES = {viii+344},
ISBN = {3-540-61788-4},
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. 72-86, 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 = {72--86},
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. 49-62, 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={49--62},
MRNUMBER={2222729},
} -
[CaSD] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, "On the product of three homogeneous linear forms and the indefinite ternary quadratic forms," Philos. Trans. Roy. Soc. London. Ser. A., vol. 248, pp. 73-96, 1955.
@article {CaSD, MRKEY = {0070653},
AUTHOR = {Cassels, J. W. S. and Swinnerton-Dyer, 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 = {73--96},
ISSN = {0080-4614},
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. 65-82, 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 = {65--82},
ISSN = {0024-6115},
MRCLASS = {10.0X},
MRNUMBER = {0041883},
ZBLNUMBER={0045.01402 },
MRREVIEWER = {R. Hull},
DOI = {10.1112/plms/s2-53.1.65},
} -
[F] H. Furstenberg, "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation," Math. Systems Theory, vol. 1, pp. 1-49, 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 = {1--49},
ISSN = {0025-5661},
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. 83-102, 1999.
@article{KleinbockBadlyAppSysJNT,
author={Kleinbock, D.},
TITLE={Badly approximable systems of affine forms},
JOURNAL={J. Number Theory},
VOLUME={79},
YEAR={1999},
PAGES={83--102},
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. 1481-1500, 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 = {1481--1500},
ISSN = {0143-3857},
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. 161-174.
@incollection {Ma, MRKEY = {1754775},
AUTHOR = {Margulis, Gregory},
TITLE = {Problems and conjectures in rigidity theory},
BOOKTITLE = {Mathematics: {{F}}rontiers and {{P}}erspectives},
PAGES = {161--174},
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. 557-570, 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 = {557--570},
ISSN = {0003-486X},
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. 235-280, 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 = {235--280},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {22E40 (22D40 28D10)},
MRNUMBER = {1106945 },
MRREVIEWER = {Gopal Prasad},
DOI = {10.1215/S0012-7094-91-06311-8},
ZBLNUMBER = {0733.22007},
} -
[Mc] C. T. McMullen, "Minkowski’s conjecture, well-rounded lattices and topological dimension," J. Amer. Math. Soc., vol. 18, iss. 3, pp. 711-734, 2005.
@article {Mc, MRKEY = {2138142},
AUTHOR = {McMullen, Curtis T.},
TITLE = {Minkowski's conjecture, well-rounded lattices and topological dimension},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {18},
YEAR = {2005},
NUMBER = {3},
PAGES = {711--734},
ISSN = {0894-0347},
MRCLASS = {11H31 (11E57 11J83 55M10)},
MRNUMBER = {2138142},
MRREVIEWER = {Stefan K{ü}hnlein},
DOI = {10.1090/S0894-0347-05-00483-2},
ZBLNUMBER = {1132.11034},
} -
[Shah91] N. A. Shah, "Uniformly distributed orbits of certain flows on homogeneous spaces," Math. Ann., vol. 289, pp. 315-334, 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={315--334},
MRNUMBER={1092178},
} -
[Sh] U. Shapira, "On a generalization of Littlewood’s conjecture," J. Mod. Dyn., vol. 3, iss. 3, pp. 457-477, 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 = {457--477},
ISSN = {1930-5311},
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. 367-392, 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={367--392},
MRNUMBER={1997950},
} -
[TsengBadlyAppSysJNT] J. Tseng, "Badly approximable affine forms and Schmidt games," J. Number Theory, vol. 129, pp. 3020-3025, 2009.
@article{TsengBadlyAppSysJNT,
author={Tseng, J.},
TITLE={Badly approximable affine forms and {S}chmidt games},
JOURNAL={J. Number Theory},
VOLUME={129},
YEAR={2009},
PAGES={3020--3025},
MRNUMBER={2560849},
}
Full Article