# Products of two Cantor sets II

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We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.

# Products of two Cantor sets I

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We consider product of two Cantor sets, and obtain the optimal estimates in terms of their thickness that guarantee that their product is an interval. This problem is motivated by the fact that the spectrum of the Labyrinth model, which is a two dimensional quasicrystal model, is given by the product of two Cantor sets. We also discuss the connection between our problem and the ”intersection of two Cantor sets” problem, which is a problem considered in several papers before.

# Diophantine approximation and bounded orbits of mixing flows on homogeneous spaces

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We sketch a proof of a theorem due to Kleinbock, and generalizing previous work of Dani and of Margulis and Kleinbock, regarding the size of the set of bounded orbits of a mixing flow on a homogeneous space. We then discuss connections to number theory, specifically the fact, proved in the same paper of Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.

# Mixing Flows on Homogeneous Spaces

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We will lay the groundwork needed to discuss some results that use homogeneous dynamics to bound the Hausdorff dimension of sets arising in number theory. Specifically, we will define mixing flows, Lie groups and algebras, homogeneous spaces, and expanding horospherical subgroups, and illustrate these concepts with a few basic examples.