## Approximation by Algebraic NumbersAlgebraic numbers can approximate and classify any real number. Here, the author gathers together results about such approximations and classifications. Written for a broad audience, the book is accessible and self-contained, with complete and detailed proofs. Starting from continued fractions and Khintchine's theorem, Bugeaud introduces a variety of techniques, ranging from explicit constructions to metric number theory, including the theory of Hausdorff dimension. So armed, the reader is led to such celebrated advanced results as the proof of Mahler's conjecture on S-numbers, the Jarnik–Besicovitch theorem, and the existence of T-numbers. Brief consideration is given both to the p-adic and the formal power series cases. Thus the book can be used for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the rich and comprehensive list of more than 600 references. |

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### Contents

1 | |

2 Approximation to algebraic numbers | 27 |

3 The classifications of Mahler and Koksma | 41 |

4 Mahlers Conjecture on Snumbers | 74 |

5 Hausdorff dimension of exceptional sets | 90 |

6 Deeper results on the measureof exceptional sets | 122 |

7 On T numbers and Unumbers | 139 |

8 Other classifications of real andcomplex numbers | 166 |

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### Common terms and phrases

Acta Arith Akad algebraic integers algebraically dependent analogue of Theorem assertion badly approximable Baker Beresnevich Bernik bers Bugeaud Cantor set Chapter 9 Cited in Chapter complex number Conjecture continued fraction converges coprime deﬁned deﬁnition denote dimension function Diophantine approximation Dodson equal established Exercise exist inﬁnitely exists a positive exists a real ﬁnite ﬁrst follows Furthermore Hausdorff dimension Hausdorff measure height implies inequality infinitely many integer integer polynomials P(X integers q interval irrational number Koksma Lebesgue measure Lemma lim sup Liouville numbers Littlewood Conjecture lower bound Mahler Mahler measure Math metric minimal polynomial Navuk non-increasing non-zero integer Number Theory numbers of degree ofdegree partial quotients positive constant positive integer positive real number PROBLEM proof of Theorem Proposition proved rational numbers real algebraic number real transcendental number resp result root Russian satisﬁes satisfying Section sequence set of real Sprindˇzuk T-numbers Theorem 1.1 U-numbers