Lior Fishman

THE SET OF BADLY APPROXIMABLE VECTORS IS STRONGLY C 1 INCOMPRESSIBLE

We prove that the countable intersection of C 1 -diffeomorphic images of cer- tain 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 Schmidts (�,�)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automati- cally 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. ∞ \ i=1 f −1 i (S)

Schmidt’s game, fractals, and orbits of toral endomorphisms

Given an integer matrix M ∈GL n (ℝ) and a point y ∈ℝ n /ℤ n , consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y ∈ℚ n /ℤ n , the set has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M ∈GL n (ℝ)∩M n × n (ℤ) and y ∈ℝ n /ℤ n , and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m × n matrices. Furthermore, we show that sets of the form and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝ n . As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint , arXiv:0912.2445] on badly approximable systems of affine forms.

INTRINSIC APPROXIMATION FOR FRACTALS DEFINED BY RATIONAL ITERATED FUNCTION SYSTEMS - MAHLER'S RESEARCH SUGGESTION

Following K. Mahlers suggestion for further research on intrinsic approxi- mation on the Cantor ternary set, we obtain a Dirichlet type theorem for the limit sets of rational iterated function systems. We further investigate the behavior of these ap- proximation functions under random translations. We connect the information regarding the distribution of rationals on the limit set encoded in the system to the distribution of rationals in reduced form by proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we prove a theorem motivating a conjecture regarding the distribution of rationals in reduced form on the Cantor ternary set.

Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces

We prove an analogue of a theorem of Pollington and Velani (Sel Math (N.S.) 11:297–307, 2005), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof incorporates the framework of intrinsic approximation on such hypersurfaces first developed in the authors’ joint work with Kleinbock (Intrinsic Diophantine approximation on manifolds, 2014. arXiv:1405.7650v2) with ideas from work of Kleinbock et al. (Sel Math (N.S.) 10:479–523, 2004).