The set of badly approximable vectors is strongly C1 incompressible

Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, Barak Weiss

Research output: Contribution to journalArticlepeer-review


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 â, 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,Î)-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.

Original languageEnglish
Pages (from-to)319-339
Number of pages21
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number2
StatePublished - Sep 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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