Approssimazione diofantea, frazioni continue e misure d’irrazionalità

Viola, Carlo

Bollettino della Unione matematica italiana, Série 8, 7A (2004) no. 2, pp. 291-320

Résumé

The diophantine approximation to a given irrational number $\alpha$ can be viewed as the quest for positive integers $s$ such that the distance from $s\alpha$ to the set of integers is exceptionally small. Thus, denoting by $r$ the integer nearest to $s\alpha$, one seeks positive integers $s$ such that $|s\alpha - r| = s|\alpha - r/s|$ is small; or, in other words, rational approximations $r/s$ to $\alpha$ for which the distance $|\alpha - r/s|$ is small but the denominator $s$ is not too large. In this paper we recall some basic facts in diophantine approximation such as the notion of best approximation, and we discuss the relation between the best approximations to $\alpha$ and the continued fraction expansion of $\alpha$. We also recall the notion of irrationality measure of $\alpha$, and we discuss some classical results about the diophantine approximation to algebraic irrational numbers, with applications to the construction of transcendental numbers (Liouville) and to the solutions of diophantine equations (Thue).