Given an $m$-dimensional weight $\mathbf{r}$ and an $n$-dimensional weight $\mathbf{s}$, we prove that the set of $(\mathbf{r};\mathbf{s})$-badly approximable $m\times n$ matrices is hyperplane absolute winning on $\mathbb{R}^{m\times n}$. This fully answers a question \cite[Question 8.2 (iii)]{Kl} of D. Kleinbock in 1...
In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in Bandi-Fregoli-Kleinbock, that is, the set of pairs of real numbers $(x,y)$ satisfying $$ \limsup_{Q\to +\infty}\ Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning....
In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/\Gamma=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\math...
In this paper, we prove that the set of counterexamples to uniform Littlewood's conjecture proposed in \cite{BFK25}, that is, the set of real pairs $(x,y)$ satisfying $$\limsup_{Q\to+\infty}\ Q\min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0$$ is hyperplane absolute winning. In particular, it has full Hausdorff...
Cheng-xun Wu, Bohan Yang· 0 citations
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