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Bad(w) is hyperplane absolute winning

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JournalGeometric And Functional Analysis
DateAccepted/In press - 8 Dec 2020
DateE-pub ahead of print (current) - 13 Mar 2021
Number of pages33
Early online date13/03/21
Original languageEnglish

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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