Metrical musings on Littlewood and friends

Haynes, A.; Jensen, J.; Kristensen, S.

doi:10.1090/S0002-9939-2013-11921-0

Metrical musings on Littlewood and friends
HTML articles powered by AMS MathViewer

by A. Haynes, J. L. Jensen and S. Kristensen PDF

Proc. Amer. Math. Soc. 142 (2014), 457-466 Request permission

Abstract:

We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teulié, and a hybrid between a conjecture of Cassels and the Littlewood conjecture. It is shown that the set of numbers satisfying a strong version of all of these conjectures is large in the sense of Hausdorff dimension restricted to the set of badly approximable numbers.

References

Boris Adamczewski and Yann Bugeaud, On the Littlewood conjecture in simultaneous Diophantine approximation, J. London Math. Soc. (2) 73 (2006), no. 2, 355–366. MR 2225491, DOI 10.1112/S0024610706022617
R. C. Baker, Metric number theory and the large sieve, J. London Math. Soc. (2) 24 (1981), no. 1, 34–40. MR 623668, DOI 10.1112/jlms/s2-24.1.34
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 248 (1955), 73–96. MR 70653, DOI 10.1098/rsta.1955.0010
Bernard de Mathan and Olivier Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), no. 3, 229–245 (French, with English summary). MR 2103807, DOI 10.1007/s00605-003-0199-y
Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513–560. MR 2247967, DOI 10.4007/annals.2006.164.513
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
Glyn Harman, Metric number theory, London Mathematical Society Monographs. New Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558
Stephen Harrap and Alan Haynes, The mixed Littlewood conjecture for pseudo-absolute values, Math. Ann. 357 (2013), no. 3, 941–960. MR 3118619, DOI 10.1007/s00208-013-0928-z
V. Jarník, Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.–Fiz. (1928–9), 91–106.
R. Kaufman, Continued fractions and Fourier transforms, Mathematika 27 (1980), no. 2, 262–267 (1981). MR 610711, DOI 10.1112/S0025579300010147
Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
Andrew D. Pollington and Sanju L. Velani, On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture, Acta Math. 185 (2000), no. 2, 287–306. MR 1819996, DOI 10.1007/BF02392812
Uri Shapira, A solution to a problem of Cassels and Diophantine properties of cubic numbers, Ann. of Math. (2) 173 (2011), no. 1, 543–557. MR 2753608, DOI 10.4007/annals.2011.173.1.11