Diophantine approximation for products of linear maps — logarithmic improvements
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- by Alexander Gorodnik and Pankaj Vishe PDF
- Trans. Amer. Math. Soc. 370 (2018), 487-507 Request permission
Abstract:
This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.References
- Alan Baker, Transcendental number theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. MR 1074572
- Dzmitry Badziahin and Sanju Velani, Multiplicatively badly approximable numbers and generalised Cantor sets, Adv. Math. 228 (2011), no. 5, 2766–2796. MR 2838058, DOI 10.1016/j.aim.2011.06.041
- Daniel Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532. MR 716835, DOI 10.1090/S0002-9947-1983-0716835-6
- J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Hafner Publishing Co., New York, 1972. Facsimile reprint of the 1957 edition. MR 0349591
- H. Davenport, Indefinite binary quadratic forms, and Euclid’s algorithm in real quadratic fields, Proc. London Math. Soc. (2) 53 (1951), 65–82. MR 41883, DOI 10.1112/plms/s2-53.1.65
- 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
- H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Theory of Computing Systems Vol.
- P. Gallagher, Metric simultaneous diophantine approximation, J. London Math. Soc. 37 (1962), 387–390. MR 157939, DOI 10.1112/jlms/s1-37.1.387
- A. Gorodnik and P. Vishe, Mixed inhomogeneous Littlewood conjecture and quantitative improvements, in preparation.
- Stephen Harrap, Twisted inhomogeneous Diophantine approximation and badly approximable sets, Acta Arith. 151 (2012), no. 1, 55–82. MR 2853045, DOI 10.4064/aa151-1-5
- A. Haynes, J. L. Jensen, and S. Kristensen, Metrical musings on Littlewood and friends, Proc. Amer. Math. Soc. 142 (2014), no. 2, 457–466. MR 3133988, DOI 10.1090/S0002-9939-2013-11921-0
- D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), no. 3, 451–494. MR 1719827, DOI 10.1007/s002220050350
- Elon Lindenstrauss and Barak Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1481–1500. MR 1855843, DOI 10.1017/S0143385701001717
- L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197–201. MR 122772, DOI 10.1090/S0002-9904-1961-10565-X
- 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
- Dennis Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982), no. 3-4, 215–237. MR 688349, DOI 10.1007/BF02392354
- Zhiren Wang, Quantitative density under higher rank abelian algebraic toral actions, Int. Math. Res. Not. IMRN 16 (2011), 3744–3821. MR 2824843, DOI 10.1093/imrn/rnq222
Additional Information
- Alexander Gorodnik
- Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1SD, United Kingdom
- Email: a.gorodnik@bristol.ac.uk
- Pankaj Vishe
- Affiliation: Department of Mathematics, Durham University, Durham DH1 3LE, United Kingdom
- MR Author ID: 1005529
- Email: pankaj.vishe@durman.ac.uk
- Received by editor(s): January 14, 2016
- Received by editor(s) in revised form: April 1, 2016, and April 7, 2016
- Published electronically: June 21, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 487-507
- MSC (2010): Primary 11D75, 11J20, 11K60, 37A17, 37A45
- DOI: https://doi.org/10.1090/tran/6953
- MathSciNet review: 3717987