The set of badly approximable vectors is strongly C¹ incompressible

Abstract

Abstract. We prove that the countable intersection of C 1 -diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in R d , improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α, β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

Bounded orbits of certain diagonalizable flows on $SL_{n}(R)/SL_{n}(Z)$

“…is winning for his game for any d ∈ N. They are also proved to be HAW in [6]. Recently, An [2] proved that Bad(r) are winning sets for Schmidt's game for any 2-weight r. The HAW property is also established for such sets by Nesharim and Simmons [17].…”

“…We have the following lemma collecting the basic properties of β-HAW subsets and HAW subsets of R d ( [6], [14], [11]): Lemma 2.1.…”

“…Note that for u ∈ Ω, the set In view of the correspondence presented in [8,Theorem 2.20], the set {x ∈ R d : u x Γ ∈ E(F + )} coincides with the set of badly approximable vectors Bad d , which is proved to be HAW already in [6]. Then we omit the remaining part of the proof here, since it is similar to the proof of the above case r 1 > r d .…”

“…When the metric space is a Riemannian manifold, α-winning sets are thick and stable with respect to countable intersections. In this paper, we will use a variant of Schmidt's (α, β)-game, i.e., the hyperplane absolute game introduced in [6] and [14]. This game has the advantage that it can be naturally defined on a differential manifold without picking a Riemannian metric while the hyperplane absolute winning (abbreviated as HAW) sets also enjoy the thickness and countable intersection properties.…”

Badly approximable points on self-affine sponges and the lower Assouad dimension

“…Since BA d has full dimension, one expects its intersection with any fractal set J ⊆ R d to have the same dimension as J , and this can be proven for certain broad classes of fractal sets J , see e.g. [4,6,12,20].…”