Diophantine approximation and Diophantine equations
نویسنده
چکیده
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦10). Next we consider negative results for rational approximation, with Liouville’s estimate for the approximation of a real algebraic number by rational numbers. We state explicit versions of Liouville’s inequality (see [5] §3.5 and exercise 3.6; [6] Course N◦3 §4.1 Lemma 24 and Proposition 26 and Course N◦4 §4.1.2), involving the absolute logarithmic height ([5] §3.2).
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