For a real number $x$, we denote $$ \|x\|=\inf_{m\in {\Bbb Z}}|x+m|.$$
Problem 1:
Roth's theorem states that given any irrational algebraic number $\alpha$ and for any $\epsilon>0$, there exists a constant $C(\alpha,\epsilon)$ such that
$$\|q\alpha\|>\frac{C(\alpha,\epsilon)}{q^{1+\epsilon}}$$ for each positive integer $q$.
Is there an explicit bound for $C(\alpha,\epsilon)$$\ ?$
Problem 2:
Let $t=(t_1,\cdots, t_k)$ be a vector whose components are all integers and do not vanish simultaneously. Suppose that $1,\alpha_1,\cdots, \alpha_k$ are real algebraic numbers and linear independent over ${\Bbb Z}$. Does there exist a constant $C$ independent of $t$ such that $$\|t\cdot\alpha\|=\|t_1\alpha_1+\cdots+t_k\alpha_k\|\geq \frac{C(\alpha,\epsilon)}{\Pi_{t_j\not=0}|t_j|^{1+\epsilon}}\ ?$$
For both of the problems, if the general case looks a little bit difficult, what about the special one that $\epsilon=1\ ?$
