Related papers: Badly approximable vectors on rational quadratic v…

We show that points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of W. M. Schmidt. As a consequence, we obtain a number field version of Schmidt's conjecture.

We prove that the set of (r_1,r_2,..,r_{d})-badly approximable vectors is a winning set if r_1=r_2=...=r_{d-1}\geq r_{d}.

We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly…

We consider approximation of vectors $\mathbf{z}\in F\otimes\mathbb{R}\cong\mathbb{R}^r\times\mathbb{C}^s$ by elements of a number field $F$ and construct examples of badly approximable vectors. These examples come from compact subspaces of…

In 1998 Kleinbock conjectured that any set of weighted badly approximable $d\times n$ real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in $\mathbf{R}^d$ in…

Let $\boldsymbol{\tau}=(\tau_1,\dots,\tau_m)\in \mathbb{R}_{\ge 0}^m$ satisfy $\sum_{i=1}^m \tau_i>1$ and $\tau_1\ge \cdots \ge \tau_m$ Let $\Psi_{\boldsymbol\tau}=(\psi_1,\dots,\psi_m)$ be given by $$ \psi_i(q)=q^{-\tau_i}, \qquad…

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier…

We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate. This implies in particular that Kurzweil's Theorem cannot be…

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 set of badly approximable systems of affine forms…

We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with…

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important…

We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of two-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt's game. As a consequence, we give a direct proof of Schmidt's conjecture using his…

A pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\textit{infinitely badly approximable}$ if \[ \liminf_{\mathbf{q}\in\mathbb{Z}^n, \|\mathbf{q}\|\to\infty}…

For $i, j > 0, i + j = 1$, the set of badly approximable vectors with weight $(i, j)$ is defined by $Bad(i, j) = \{(x, y) \in \R^2 : \exists c > 0 \forall q\in\N, \;\; \max\{q||qx||^{1/i}, q||qy||^{1/j} \} > c\}$, where $||x||$ is the…

The set of badly approximable numbers, Bad, is known to be winning for Schmidt's game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the…

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 a analogue of Schmidt's…

In this paper we investigate the problem of how well points in finite dimensional p-adic solenoids can be approximated by rationals. The setting we work in was previously studied by Palmer, who proved analogues of Dirichlet's theorem and…

Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this game (and modifications of it) have been applied to many examples of complete metric spaces (X, d) to show that the set of "badly approximable points", with respect…

We show an analogue of a theorem of An, Ghosh, Guan, and Ly on weighted badly approximable vectors for totally imaginary number fields. We show that for $G=\mathrm{SL}_2(\mathbb{C})\times\dots\times\mathrm{SL}_2(\mathbb{C})$ and $\Gamma<G$…

Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$ \{\xi =(\xi_1,...,\xi_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||q\xi_i|| \to \infty,\,\,\,\, q\to \infty\} $$ is an…