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.

1998年，克莱因博克（Kleinbock）推测，就施密特的游戏而言，任何一组加权的非常近似的逼近\（d \ times \ n）实数矩阵都是获胜的子集。在本文中，我们通过证明相应的加权严重近似向量集是超平面绝对获胜，充分证明了\（{{mathbb {R}} ^ d \）中向量的猜想。该证明使用在Ahlfors常规绝对衰减测度和Kleinbock，Lindenstrauss和Weiss引起的一类分形测度的定量不偏离估计的支持下进行的Cantor势博弈。为了确定Cantor潜在博弈中相关获胜策略的存在，我们引入了一种新方法，该方法使用对晶格空间的两个独立对角线动作。