Bad(w) is hyperplane absolute winning

Victor Beresnevich, Erez Nesharim, Lei Yang

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)1-33
JournalGeometric And Functional Analysis
Publication statusPublished - 13 Mar 2021

Bibliographical note

© 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

Cite this