# More about Roth's theorem: bound for the constant and multidimensional case

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 $$\epsilon=1\ ?$$

## 1 Answer

Prof. Lilu Zhao of Shandong University informed about the follwing paper

Simultaneous approximation to algebraic numbers by rationals

in which Theorem 2 solves the question affirmatively.