Badly approximable systems of affine forms and incompressibility on fractals

JOURNAL OF NUMBER THEORY

卷:133

Article

Broderick, Ryan¹  Fishman, Lior²  Simmons, David² 

[1] Northwestern Univ, Dept Math, Evanston, IL 60208 USA

[2] Univ N Texas, Dept Math, Denton, TX 76203 USA

关键词: Diophantine approximation;    Linear and affine forms;    Badly approximable;    Geometric measure theory;    Fractals;    Schmidt's game;   

DOI  :  10.1016/j.jnt.2012.12.004

来源: Elsevier

【 摘 要 】

We explore and refine techniques for estimating the Hausdorff dimension of Diophantine exceptional sets and their diffeomorphic images. This work is directly motivated by a recent advance in geometric measure theory, which facilitates the use of games in bounding the dimension of a set's intersection with a sufficiently regular fractal. Specifically, we use a variant of Schmidt's game to deduce the strong C-1 incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free