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We shall say that $\xi\in\mathbb{R}$ is approximable by rationals to order n if there is a $K(\xi)$ for which $$|\frac pq -\xi|<\frac {K(\xi)}{q^n}$$ has infinitly many solutions. (according to the book: THE THEORY OF NUMBERS by Hardy & Wright)

While reading about diophantine approximation I've encountered the two following formulations of Liouville's approximation theorem:

(1) As it appeared in the classic book by Hardy & Wright, the Liouville's approximation theorem states that:

A real algebraic number of degree n is not approximable to any order greater then n

(where I interpret this statement in the sense of the definition given above)

(2) According to: http://mathworld.wolfram.com/LiouvillesApproximationTheorem.html

and also

https://en.wikipedia.org/wiki/Diophantine_approximation

The Theorem states that:

$\exists c(\xi)>0$ s.t. $$|\frac pq -\xi|\geq\frac {c(\xi)}{q^n}$$

holds for all integers p and q where q > 0. where $\xi$ is an irrational algebraic number of degree n over the rational numbers

My question is: why these two statements equivalent?

What am I missing here?

S.H.
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1 Answers1

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After asking someone I know, I got the following answer:

These two statements aren't equal. What we use to call Liouville's Approximation Thm. is the 2nd one and it's stronger than the 1st. Here is a proof..

Claim: (2) implies (1)

Pf. Let $\xi$ be an irrational algebraic # of degree d>0 (over the rationals). Denote by $c(\xi)>0$ the mentioned constant from statement (2) .

Lets assume by contradiction that $\xi$ is approximable to order d+1, and let $K(\xi)>0$ be the real # from the definition. By the definition, there's $\frac{p}{q}$ with sufficiently large q>0 s.t. $$K(\xi)<q \cdot c(\xi)$$ and also $$|\frac{p}{q}-\xi|<\frac{K(\xi)}{q^{d+1}}<\frac{K(\xi)}{q} \cdot \frac{1}{q^d}<\frac{c(\xi)}{q^d}$$ which contradicts Liouville's Thm $\square$

S.H.
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