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.