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Diophantine approximation for products of linear maps -- logarithmic improvements

Authors: Alexander Gorodnik and Pankaj Vishe
Journal: Trans. Amer. Math. Soc. 370 (2018), 487-507
MSC (2010): Primary 11D75, 11J20, 11K60, 37A17, 37A45
Published electronically: June 21, 2017
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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.

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  • [1] Alan Baker, Transcendental number theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. MR 1074572
  • [2] Dzmitry Badziahin and Sanju Velani, Multiplicatively badly approximable numbers and generalised Cantor sets, Adv. Math. 228 (2011), no. 5, 2766-2796. MR 2838058,
  • [3] Daniel Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), no. 2, 509-532. MR 716835,
  • [4] J. W. S. Cassels, An introduction to Diophantine approximation, Hafner Publishing Co., New York, 1972. MR 0349591
  • [5] H. Davenport, Indefinite binary quadratic forms, and Euclid's algorithm in real quadratic fields, Proc. London Math. Soc. (2) 53 (1951), 65-82. MR 0041883,
  • [6] Bernard de Mathan and Olivier Teulié, Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), no. 3, 229-245. MR 2103807,
  • [7] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Theory of Computing Systems Vol.
  • [8] P. Gallagher, Metric simultaneous diophantine approximation, J. London Math. Soc. 37 (1962), 387-390. MR 0157939,
  • [9] A. Gorodnik and P. Vishe, Mixed inhomogeneous Littlewood conjecture and quantitative improvements, in preparation.
  • [10] Stephen Harrap, Twisted inhomogeneous Diophantine approximation and badly approximable sets, Acta Arith. 151 (2012), no. 1, 55-82. MR 2853045,
  • [11] 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,
  • [12] D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), no. 3, 451-494. MR 1719827,
  • [13] 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,
  • [14] L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197-201. MR 0122772,
  • [15] 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,
  • [16] 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,
  • [17] 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,
  • [18] Zhiren Wang, Quantitative density under higher rank abelian algebraic toral actions, Int. Math. Res. Not. IMRN 16 (2011), 3744-3821. MR 2824843,

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Additional Information

Alexander Gorodnik
Affiliation: School of Mathematics, University of Bristol, Bristol BS8 1SD, United Kingdom

Pankaj Vishe
Affiliation: Department of Mathematics, Durham University, Durham DH1 3LE, United Kingdom

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
Article copyright: © Copyright 2017 American Mathematical Society

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