论文标题
在乘以近似值的倍数上
On Multiplicatively Badly Approximable Vectors
论文作者
论文摘要
令$ \ langle x \ rangle $表示$ x \ in \ mathbb {r} $的距离到整数$ \ mathbb {z} $。 Littlewood的猜想指出，所有对$（α，β）\ in \ Mathbb {r}^{2} $ the产品$ q \ langle q \ langleqα\ rangle \ langleqβ\ rangle $获得任意接近$ 0 $ q \ in \ intbb in \ mathbb的$ 0 $ q \ n n n n n n} $ nimes n ifts n n Infty ty n iff。 Badziahin表明，如果一个因子$ \ log q \ cdot \ log \ log \ log q $已添加到产品中，则相同的语句变为false。在本文中，我们将Badziahin的结果推广为向量$ \BoldSymbolα\ in \ Mathbb {r}^{d} $，更换函数$ \ log q \ cd \ cdot \ cdot \ cdot \ log \ log q $ by $（\ log q） $ d = 2 $。我们的方法是基于著名的DANI对应关系的新版本，而在晶格空间上，尤其是适合于理性近似产物的研究。我们认为，这种信件具有独立的利益。
Let $\langle x\rangle$ denote the distance from $x\in\mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(α,β)\in\mathbb{R}^{2}$ the product $q\langle qα\rangle\langle qβ\rangle$ attains values arbitrarily close to $0$ as $q\in\mathbb{N}$ tends to infinity. Badziahin showed that if a factor $\log q\cdot \log\log q$ is added to the product, the same statement becomes false. In this paper, we generalise Badziahin's result to vectors $\boldsymbolα\in\mathbb{R}^{d}$, replacing the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for any $d\geq 2$, and thereby obtaining a new proof in the case $d=2$. Our approach is based on a new version of the well-known Dani Correspondence between Diophantine approximation and dynamics on the space of lattices, especially adapted to the study of products of rational approximations. We believe that this correspondence is of independent interest.
