Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $1,a_1,...,a_k$ are linearly independent over $\mathbb{Q}$ and such that $(a_1,...,a_k)$ is not "badly approximable"?
Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $$[\mathbb{Q}(a_1,...,a_k):\mathbb{Q}]>k+1$$ but such that $(a_1,...,a_k)$ is "badly approximable"?
knowing whether either of these is open would be a great help.
By badly approximable I mean that there exists $c>0$ such that the $\ell_\infty$ distance from $(qa_1,...,qa_k)$ to $\mathbb{Z}^k$ is greater than $cq^{-1/k}$ for all large $q\in \mathbb{N}$.
Or the equivalent notion that there exists $c>0$ such that the distance from $\sum_i q_i a_i$ to $\mathbb{Z}$ is greater than $c\|q\|_{\ell_\infty}^{-k}$ for all large $q\in \mathbb{Z}^k$.
