Abstract: Khintchine's theorem in Diophantine approximation gives a zero one law describing the approximability of typical points by rational points. In 1984, Mahler asked how well points on the middle third Cantor set can be approximated. His question fits into an attempt to determine conditions under which subsets of Euclidean space inherit the Diophantine properties of the ambient space.
In this talk, we discuss a complete analogue of Khintchine’s theorem for certain fractal measures. Our results hold for fractals generated by rational similarities of Euclidean space that have sufficiently small Hausdorff co-dimension. The main ingredient to the proof is an effective equidistribution theorem for associated fractal measures on the space of unimodular lattices. The latter is established using a spectral gap property of a type of Markov operators associated with a random walk related to the generating similarities. This is joint work with Osama Khalil.


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Meeting ID: 851 5954 2997
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