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Badly approximable points in twisted Diophantine approximation and Hausdorff dimension

Tom 177 / 2017

Paloma Bengoechea, Nikolay Moshchevitin Acta Arithmetica 177 (2017), 301-314 MSC: 11K60, 11J83, 11J20. DOI: 10.4064/aa8234-11-2016 Opublikowany online: 22 February 2017

Streszczenie

For any with \sum_{i=1}^nj_i=1 and any \theta\in\mathbb R^n, let {\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)} denote the set of points \eta\in\mathbb R^n for which \max_{1\leq i\leq n}(\|q\theta_i-\eta_i\|^{1/j_i}) \gt c/q for some positive constant c=c(\eta) and all q\in\mathbb N. These sets are the ‘twisted’ inhomogeneous analogue of \mathrm{Bad}(j_1,\ldots,j_n) in the theory of simultaneous Diophantine approximation. It has been shown that they have full Hausdorff dimension in the non-weighted setting, i.e. provided that j_i=1/n, and in the weighted setting when \theta is chosen from \mathrm{Bad}(j_1,\ldots,j_n). We generalise these results by proving the full Hausdorff dimension in the weighted setting without any condition on \theta. Moreover, we prove \dim(\mathrm{Bad}_{\theta}(j_1,\ldots,j_n)\cap\mathrm{Bad}(1,0,\ldots,0)\cap\cdots\cap\mathrm{Bad}(0,\ldots,0,1))=n.

Autorzy

  • Paloma BengoecheaDepartment of Mathematics
    ETH Zürich
    Ramistrasse 101
    8092 Zürich, Switzerland
    e-mail
  • Nikolay MoshchevitinFaculty of Mathematics and Mechanics
    Moscow State University
    Leninskie Gory 1
    GZ MGU, 119991 Moscow, Russia
    e-mail

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