论文标题
糟糕（$ \ mathbf {w} $）是超平面绝对获胜
Bad($\mathbf{w}$) is hyperplane absolute winning
论文作者
论文摘要
1998年，克莱恩博克（Kleinbock）猜想，在施密特（Schmidt）的游戏意义上，任何一套加权差不多的$ d \ times n $ n $ real矩阵都是胜利的子集。在本文中，我们通过证明相应的加权不良近似矢量的相应集合是超平面绝对赢得的，以$ \ mathbf {r}^d $中的向量证明了这个猜想。该证明使用了在AHLFORS常规衰减措施的支持下进行的Cantor潜在游戏，以及由于Kleinbock，Lindenstrauss和Weiss而导致的一类分形措施的定量非差异估计。为了在Cantor潜在游戏中建立相关的获胜策略的存在，我们使用在格子空间上使用两种独立的对角线动作引入了一种新方法。
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 $\mathbf{R}^d$ 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 non-divergence 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.
