Beresnevich, Victor orcid.org/0000-0002-1811-9697, Nesharim, Erez and Yang, Lei (2021) Bad(w) is hyperplane absolute winning. Geometric And Functional Analysis. pp. 1-33. ISSN 1420-8970
Abstract
In 1998 Kleinbock conjectured that any set of weighted badly approximable $d\times n$ real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in $\mathbb{R}^d$ in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential game played on the support of Ahlfors regular absolutely decaying measures and the quantitative nondivergence estimate for a class of fractal measures due to Kleinbock, Lindenstrauss and Weiss. To establish the existence of a relevant winning strategy in the Cantor potential game we introduce a new approach using two independent diagonal actions on the space of lattices.
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Copyright, Publisher and Additional Information: | © 2021 Springer Nature Switzerland AG. Part of Springer Nature. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details |
Dates: |
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Institution: | The University of York |
Academic Units: | The University of York > Faculty of Sciences (York) > Mathematics (York) |
Depositing User: | Pure (York) |
Date Deposited: | 09 Dec 2020 10:20 |
Last Modified: | 01 Feb 2024 00:39 |
Published Version: | https://doi.org/10.1007/s00039-021-00555-7 |
Status: | Published |
Refereed: | Yes |
Identification Number: | https://doi.org/10.1007/s00039-021-00555-7 |
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