论文标题
在加权奇异向量的Hausdorff尺寸的下限
On a lower bound of Hausdorff dimension of weighted singular vectors
论文作者
论文摘要
令$ w =（w_1，\ dots，w_d）$为$ d $ - 阳性实际数字的总计，以便$ \ sum_ {i} w_i = 1 $和$ w_1 \ geq \ geq \ cdots \ geq w_d $。 a $ d $ -dimensional vector $ x =（x_1，\ dots，x_d）\ in \ mathbb {r}^d $ in the $ w $ -singular如果每一个$ε> 0 $都存在$ t_0> 1 $，则$ t_0> 1 $，以便所有$ t> t_0 $ t_0 $ t_0 $ t_ t_0 $ quq iq fe \ max_ [\ max_ [\ max_] d} | qx_i -p_i |^{\ frac {1} {w_i}} <\fracε{t} \ quad \ quad \ text {and} \ quad 0 <q <q <t \]的integer solugin $ \ mathbb {z}^d \ times \ mathbb {z} $。我们证明，$ w $ singular vectors in $ \ mathbb {r}^d $在下面以$ d- \ frac {1} {1+w_1} $界定的hausdorff尺寸。我们的结果部分扩展了Liao等人的先前结果。 [$ \ mathbb {r}^2 $，J。EUR中的加权奇异向量的Hausdorff尺寸。数学。 Soc。 22（2020），833-875]。
Let $w=(w_1,\dots,w_d)$ be a $d$-tuple of positive real numbers such that $\sum_{i}w_i =1$ and $w_1\geq \cdots \geq w_d$. A $d$-dimensional vector $x=(x_1,\dots,x_d)\in\mathbb{R}^d$ is said to be $w$-singular if for every $ε>0$ there exists $T_0>1$ such that for all $T>T_0$ the system of inequalities \[ \max_{1\leq i\leq d}|qx_i - p_i|^{\frac{1}{w_i}} < \fracε{T} \quad\text{and}\quad 0<q<T \] have an integer solution $(\mathbf{p},q)=(p_1,\dots,p_d,q)\in \mathbb{Z}^d \times \mathbb{Z}$. We prove that the Hausdorff dimension of the set of $w$-singular vectors in $\mathbb{R}^d$ is bounded below by $d-\frac{1}{1+w_1}$. Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in $\mathbb{R}^2$, J. Eur. Math. Soc. 22 (2020), 833-875].
