On Multiplicatively Badly Approximable Vectors

Let $⟨x⟩$ denote the distance from $x\in \mathbb{R}$ to the set of integers $\mathbb{Z}$. The Littlewood Conjecture states that for all pairs $(\alpha ,\beta )\in {\mathbb{R}}^2$ the product $q⟨q\alpha ⟩⟨q\beta ⟩$ 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 $\mathit{\alpha }\in {\mathbb{R}}^d$, replacing the function $\log q\cdot \log \log q$ by $(\log q{)}^{d-1}\cdot \log \log q$ for any $d\ge 2$, 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.