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The set of badly approximable vectors is strongly C1 incompressible

Published online by Cambridge University Press:  23 May 2012

RYAN BRODERICK
Affiliation:
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
LIOR FISHMAN
Affiliation:
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
DMITRY KLEINBOCK
Affiliation:
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
ASAF REICH
Affiliation:
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
BARAK WEISS
Affiliation:
Department of Mathematics, Ben Gurion University, Be'er Sheva, Israel84105 e-mail: barakw@math.bgu.ac.il

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

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