Abstract
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of \({\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}}\) are uniformly eventually bounded.
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Part of the third author’s PhD thesis at the Hebrew University of Jerusalem. The partial support of ISF grant number 1157/08, as well as the Binational Science Foundation grant 2008454 is acknowledged.
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Einsiedler, M., Fishman, L. & Shapira, U. Diophantine Approximations on Fractals. Geom. Funct. Anal. 21, 14–35 (2011). https://doi.org/10.1007/s00039-011-0111-1
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DOI: https://doi.org/10.1007/s00039-011-0111-1