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 $\epsilon=1\ ?$



More about Roth's theorem: bound for the constant and multidimensional case
Prof. Lilu Zhao of Shandong University informed about the follwing paper


[Simultaneous approximation to algebraic numbers by rationals](https://projecteuclid.org/journals/acta-mathematica/volume-125/issue-none/Simultaneous-approximation-to-algebraic-numbers-by-rationals/10.1007/BF02392334.full)


in which Theorem 2 solves the question affirmatively.


