# Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of:

(1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in lowest terms) such that $|x-(p/q)| < 1/q^n$. Can you generalize the above cited statement in more detailed manner?

(2) Let $P$= $(x_1, x_2, ..., x_n)$ be in $P^n (Q)$. Then the logarithmic height of $P$ is defined by $H(P)$= $\sum$(max{$|$$x_1$$|_p$,$|$$x_2$$|_p$,...,$|$$x_n$$|_p$}), where p is in M. How this is defined?kindly explain with good example, if necessary.

(3) Let $S$ be a projective space over a number field and our function @ to be a rational function, then through their substitution, how can we investigate the iterates of @ using the ideas and tools from both Diophantine geometry and Dynamical systems?

The all cited questions, I found in literature. Due to poor knowledge in Number Theory, especially Diophantine equations and Geometry, I could not get. Kindly explain.

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I suggest starting a new thread for each question you have, as well as detailing any thoughts you have on each of the questions. –  Clayton Jan 18 '13 at 6:12
I guess that you are reading books by Silverman - you will get better answers if you say a bit about your background. Anyway, for (1), the keyword to search is Roth's theorem. –  Sanchez Jan 18 '13 at 6:14
Maybe the thing to do is first better your knowledge of Number Theory, and come back to these questions when you are ready for them. –  Gerry Myerson Jan 18 '13 at 6:27
@Thank you for your information. –  rr282828 Jan 19 '13 at 5:04
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