The Set of Badly Approximable Vectors Is Strongly C Incompressible
نویسنده
چکیده
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 (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.
منابع مشابه
Badly Approximable Numbers and Vectors in Cantor-like Sets
We show that a large class of Cantor-like sets of Rd, d ≥ 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d ≥ 2. An analogous result is also proved for subsets of Rd arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable...
متن کاملThe Geometry of Badly Approximable Vectors
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...
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.
متن کاملBadly approximable systems of linear forms over a field of formal series
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 Jarník’s Theorem on badly approximable numbers.
متن کاملVery Badly Approximable Matrix Functions
We study in this paper very badly approximable matrix functions on the unit circle T, i.e., matrix functions Φ such that the zero function is a superoptimal approximation of Φ. The purpose of this paper is to obtain a characterization of the continuous very badly approximable functions. Our characterization is more geometric than algebraic characterizations earlier obtained in [PY1] and [AP]. I...
متن کامل