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 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.