The set of badly approximable vectors is strongly C1 incompressible

By Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, Barak Weiss. Mathematical Proceedings of the Cambridge Philosophical Society, 2011

89 citations recorded by the source corpus.

Abstract We prove that the countable intersection of C1-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in ℝd, 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0fd99c2806cafdd2f09ad6897b9411dec3659495

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Jinpeng An
2. Lior Fishman
3. O. Sarig

In Geometry and Topology

1. Jinpeng An
2. Mariusz Urba'nski
3. David Simmons
4. Lior Fishman
5. Victor Beresnevich

In Mathematical Physics

1. Jinpeng An
2. Mariusz Urba'nski
3. David Simmons
4. Lior Fishman
5. Victor Beresnevich
6. O. Sarig
7. V. Mayer

Citations in the corpus

• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (2013, background): “If Z = R and H = {affine hyperplanes in R}, then this game corresponds to the hyperplane game introduced in [14].”
• Two-dimensional badly approximable vectors and Schmidt's game (2012, background): “It should be noted that several stronger variants of Schmidt’s game have been defined and used to problems in Diophantine approximation (see, for example, [14, 3]).”