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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 $\epsilon=1\ ?$

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1 Answer 1

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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.

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