# Dimension bound for doubly badly approximable affine forms

@article{Kim2019DimensionBF, title={Dimension bound for doubly badly approximable affine forms}, author={Wooyeon Kim and Seonhee Lim}, journal={arXiv: Dynamical Systems}, year={2019} }

We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows.
It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(\epsilon)$ of $\epsilon$-badly approximable target $b$ is not full, and that for real numbers $\alpha$, $\dim_H Bad_\alpha(\epsilon)=1$ if and only if $\alpha$ is singular on average. We show that if $\dim_H Bad_A(\epsilon)=m…

## References

SHOWING 1-10 OF 11 REFERENCES

Dimension bound for badly approximable grids

- Mathematics
- 2017

We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $\epsilon>0$ there exists $\delta>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n} \ge…

On shrinking targets for Z^m actions on tori

- Mathematics
- 2008

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is…

A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

- Mathematics
- 2017

ON BADLY APPROXIMABLE NUMBERS AND CERTAIN GAMES

- Mathematics
- 1966

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…

Singular systems of linear forms and non-escape of mass in the space of lattices

- Mathematics
- 2014

Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in one-to-one correspondence with certain divergent orbits of…

Badly approximable systems of affine forms, fractals, and Schmidt games

- Mathematics
- 2009

Abstract A badly approximable system of affine forms is determined by a matrix and a vector. We show Kleinbock's conjecture for badly approximable systems of affine forms: for any fixed vector, the…

Diagonal actions on locally homogeneous spaces

- Mathematics
- 2008

Contents 1. Introduction 1 2. Ergodic theory: some background 4 3. Entropy of dynamical systems: some more background 6 4. Conditional Expectation and Martingale theorems 12 5. Countably generated…

Fractal geometry, second ed

- Mathematical foundations and applications
- 2003

Groshev, Un théorème sur les systémes des formes linéaires (a theorem on a system of linear forms)

- Dokl. Akad. Nauk SSSR
- 1938