Badly approximable systems of affine forms and incompressibility on fractals
نویسندگان
چکیده
منابع مشابه
Badly Approximable Systems of Affine Forms
We prove an inhomogeneous analogue of W.M. Schmidt’s (1969) theorem on the Hausdorff dimension of the set of badly approximable systems of linear forms. The proof is based on ideas and methods from the theory of dynamical systems, in particular, on abundance of bounded orbits of mixing flows on homogeneous spaces of Lie groups.
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For any real number θ, the set of all real numbers x for which there exists a constant c(x) > 0 such that infp∈Z |θq−x− p| ≥ c(x) |q| for all q ∈ Z\{0} is an 1/8-winning set.
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We prove that for every M,N ∈ N, if τ is a Borel, finite, absolutely friendly measure supported on a compact subset K of R , then K ∩ BA(M,N) is a winning set in Schmidt’s game sense played on K, where BA(M,N) is the set of badly approximable M × N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting simila...
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1. Introduction. A number a is called badly approximable if j a — p/q | > c/q2 for some c > 0 and all rationals pjq. It is known that an irrational number a is badly approximable if and only if the partial denominators in its continued fraction are bounded [4, Theorem 23]. In a recent paper [7] I proved results of the following type: // fuf2, • ■ • are differenliable functions whose derivatives...
متن کاملBadly approximable systems of linear forms over a field of formal series par Simon KRISTENSEN
We prove that the Hausdorff dimension of the set of badly approximable systems of m linear forms in n variables over the field of Laurent series with coefficients from a finite field is maximal. This is an analogue of Schmidt’s multi-dimensional generalisation of Jarńık’s Theorem on badly approximable numbers.
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2013
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2012.12.004