  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 $\boldsymbol{\alpha}\in\mathbb{R}^{d}$, replacing
the function $\log q\cdot \log\log q$ by $(\log q)^{d-1}\cdot\log\log q$ for
any $d\geq 2$, 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.
