Bad($\mathbf{w}$) is hyperplane absolute winning

Abstract

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.

Winning property of badly approximable points on curves

“…As we mentioned in the introduction it is currently known from [GY19] that Bad(r) is hyperplane absolute winning in the case r 1 = • • • = r n−1 ≥ r n . Further developing the ideas of this paper in [BNY20] we establish the following unconditional result. Theorem 6.1.…”

“…In particular, in [BNY20] we further develop the framework of intersections with fractals which requires suitable extensions of Lemmas 1.3 and 1.4 underlining the equivalence (1.7). Also, in [BNY20] we demonstrate an equivalent approach that uses Cantor winning sets instead of generalised Cantor sets discussed in §2.2 above.…”