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Appendix E

Diophantine approximation

In this appendix, we survey classical results on approximation to real (algebraic) numbers by rational numbers and, more generally, by algebraic numbers of bounded degree. For additional results (and proofs), the reader is directed to the monographs [111, 466, 635].

E.1 Rational approximation

DEFINITION E.1. The irrationality exponent μ(ξ) of the real number ξ is the supremum of the real numbers μ for which the inequality

has infinitely many solutions in non-zero integers p and q.

We begin with an easy result.

THEOREM E.2. Let ξ be a real number. We have μ(ξ) = 1 if ξ is rational, and μ(ξ) ≥ 2 otherwise.

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