The set of badly approximable vectors is strongly C1 incompressible
نویسندگان
چکیده
منابع مشابه
The Set of Badly Approximable Vectors Is Strongly C Incompressible
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 Rd, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt’s (α, β)-game and show that our sets are hyperplane absolute winning...
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A vector v = (v1, v2, . . . , vk) in R k is -badly approximable if for all m, and for 1 ≤ j ≤ k, the distance ‖mvj‖ from mvj to the nearest integer satisfies ‖mvj‖ > m−1/k. A badly approximable vector is a vector that is -badly approximable for some > 0. For the case of k = 1, these are just the badly approximable numbers, that is, the ones with a continued fraction expansion for which the part...
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I consider the Diophantine approximation problem of sup-norm simultaneous rational approximation with common denominator of a pair of irrational numbers, and compute explicitly some pairs with large approximation constant. One of these pairs is the most badly approximable pair yet computed. The theory of approximation of a single irrational number by rationals is well known, and for our purpose...
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ژورنال
عنوان ژورنال: Mathematical Proceedings of the Cambridge Philosophical Society
سال: 2012
ISSN: 0305-0041,1469-8064
DOI: 10.1017/s0305004112000242