Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - September 2012 - Volume 153, Issue 02  

Research Article

The set of badly approximable vectors is strongly C 1 incompressible

RYAN BRODERICKa1, LIOR FISHMANa1, DMITRY KLEINBOCKa1, ASAF REICHa1 and BARAK WEISSa2

a1 Department of Mathematics, Brandeis University, Waltham MA 02454-9110, U.S.A. e-mail: ryan@math.northwestern.edu, lfishman@unt.edu, kleinboc@brandeis.edu, azmreich@brandeis.edu

a2 Department of Mathematics, Ben Gurion University, Be'er Sheva, Israel 84105 e-mail: barakw@math.bgu.ac.il

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.

(Received June 14 2011)

(Revised February 21 2012)

(Online publication May 23 2012)