CONTINUED FRACTIONS

Contents......Page 12
\rPreface......Page 6
1. Introduction......Page 16
1.1 The Additive Subgroup of the Integers Generated by a and b......Page 19
1.2 Continuants......Page 21
1.3 The Continued Fraction Expansion of a Real Number......Page 22
1.4 Quadratic Irrationals......Page 23
1.5 The Tree of Continued Fraction Expansions......Page 27
1.6 Diophantine Approximation......Page 28
1.7 Other Known Continued Fraction Expansions......Page 34
2.1 Other gcd's......Page 38
2.2 Continued Fraction Expansions for Complex Numbers......Page 41
2.3 The Lattice Reduction Algorithm of Gauss......Page 45
3. Continued Fractions with Small Partial Quotients......Page 48
3.1 The Sequence ({na}) of Multiples of a Number......Page 54
3.2 Discrepancy......Page 60
3.3 The Sum of {na} from 1 to N......Page 61
4.1 Ergodic Maps......Page 64
4.2 Terminology......Page 65
4.3 Nair's Proof......Page 67
4.4 Generalization to EM......Page 68
4.5 A Natural Extension of the Dynamic System (EM u T)......Page 73
5.1 The Schmidt Regular Chains Algorithm......Page 82
5.2 The Hurwitz Complex Continued Fraction......Page 86
5.3 Notation......Page 88
5.4 Growth of |qn| and the Quality of the Hurwitz Approximations......Page 90
5.5 Distribution of the Remainders......Page 94
5.6 A Class of Algebraic Approximants with Atypical Hurwitz Continued Fraction Expansions......Page 97
5.7 The Gauss-Kuz'min Density for the Hurwitz Algorithm......Page 101
6. Multidimensional Diophantine Approximation......Page 114
6.1 The Hermite Approximations to a Real Number......Page 122
6.2 The Lagarias Algorithm in Higher Dimensions......Page 126
6.3 Convexity of Expansion Domains in the Lagarias Algorithm......Page 132
7.1 Introduction......Page 142
7.2 Outline and Plan of Proof......Page 145
7.3 Proof of the Existence of a Unit u E Q(a) of Degree n......Page 147
7.4 The Sequence v[k] of Units with Comparable Conjugates......Page 148
7.5 Good Units and Good Denominators......Page 150
7.6 Ratios of Consecutive Good q......Page 152
7.7 The Surfaces Associated With the Scaled Errors......Page 153
7.8 The General Case of Algebraic Numbers in Q(a)......Page 156
8.1 The Binary Trees of EN......Page 160
8.2 Sums of Bridges Covering [aN wN]......Page 165
8.3 The Lagrange and Markoff Spectra......Page 168
9. Functional-Analytic Techniques......Page 170
9.1 Continued Fraction Cantor Sets......Page 176
9.2 Spaces and Operators......Page 182
9.3 Positive Operators......Page 191
9.4 An Integral Representation of gM a......Page 192
9.5 A Hilbert Space Structure for C when s = o is Real......Page 196
9.6 The Uniform Spectral Gap......Page 205
9.7 Log Convexity of AM......Page 210
10. The Generating Function Method......Page 220
10.1 Entropy......Page 221
10.2 Notation......Page 222
10.3 A Sampling of Results......Page 223
11. Conformal Iterated Function Systems......Page 228
12. Convergence of Continued Fractions......Page 232
12.1 Some General Results and Techniques......Page 233
12.2 Special Analytic Continued Fractions......Page 239
Bibliogrpahy......Page 248
Index......Page 256