Badw is hyperplane absolute winning

In 1998 Kleinbock conjectured that any set of weighted badly approximable d × 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 R d in arbitrary dimensions by showing that the corresponding set of weighted badly approxim...

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Published inGeometric and functional analysis Vol. 31; no. 1; pp. 1 - 33
Main Authors Beresnevich, Victor, Nesharim, Erez, Yang, Lei
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2021
Springer Nature B.V
Subjects
Analysis
Games
Hyperplanes
Lattices
Mathematics
Mathematics and Statistics
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Summary:In 1998 Kleinbock conjectured that any set of weighted badly approximable d × 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 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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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-021-00555-7