Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics

Foreword......Page 6
Preface......Page 8
Contents......Page 11
Contributors......Page 13
 References......Page 15
  1.1 Historical Notes......Page 17
  1.2 The von Mangoldt Function......Page 20
  1.3 The Tchebychev Inequalities......Page 21
  1.4 The Mertens Theorems......Page 23
  Reference......Page 24
 2 Arithmetic Functions......Page 25
 3 Dirichlet Series......Page 31
 4 Euler's Gamma Function......Page 38
  References......Page 39
  5.1 The Functional Equation of Zeta......Page 40
  5.2 Entire Functions of Finite Order......Page 44
  5.3 Infinite Product for ξ(s)......Page 45
  5.4 A Zero Free Region for Zeta......Page 49
  5.5 The Number of Zeros of Zeta......Page 51
  5.6 An Explicit Formula for ψ(x)......Page 56
  5.7 The Prime Numbers Theorem......Page 62
  References......Page 65
  6.1 Analytic Form of the Large Sieve......Page 66
  6.2 The Beurling–Selberg Function......Page 69
  6.3 The Sieve......Page 72
  6.4 The Arithmetic Form of the Large Sieve......Page 73
  6.5 Twin Prime Numbers......Page 78
  6.6 Sums of Characters......Page 85
  6.7 Complements......Page 86
  6.8 Double Large Sieve......Page 87
  References......Page 91
  7.1 The Circle Method......Page 92
  7.2 Major Arcs......Page 95
  7.3 Minor Arcs......Page 101
  7.4 The Vaughan Identity......Page 103
  7.5 Sums of Type I......Page 105
  7.6 Sums of Type II......Page 106
  7.8 Combinatorial Identities......Page 109
  8.1 Uniformly Distributed Sequences Modulo 1......Page 111
  8.2 Upper Bounds on Exponential Sums......Page 115
   Complement: Generalized van der Corput Inequality......Page 124
  References......Page 125
Part II Interactions Between Arithmetic and Dynamics......Page 126
  9.1 Introduction......Page 127
  9.2 General Setting and Notions......Page 128
   9.3.1 Trace Maps......Page 130
   9.3.2 Toral Automorphisms......Page 131
   9.3.3 Polynomial Automorphisms of the Plane......Page 133
  9.4 Shift Spaces with Faithful Z-action......Page 135
  9.5 Shift Spaces with Faithful Zd-action......Page 138
   9.5.1 Tiling Dynamical Systems as Subshifts......Page 139
   9.5.2 Shifts of Algebraic Origin......Page 140
  References......Page 143
  10.1 Introduction to Homogeneous Dynamics......Page 146
   10.1.1 The Space of Lattices......Page 149
   10.1.2 The Modular Surface......Page 152
   10.2.1 Kloosterman Sums......Page 155
   10.2.2 A Disjointness Result......Page 156
   10.2.3 The Connection......Page 157
  10.3 Kloosterman Sums and Spectral Gap......Page 160
   10.3.1 Ineffective Equidistribution......Page 161
  10.4 Sparse Equidistribution of Primitive Points......Page 163
   10.4.1 The Discrete Periodic Horocycle Orbit: Ineffective Discussion......Page 164
   10.4.2 The Discrete Periodic Horocycle: Effective Discussion......Page 165
   10.4.3 Selecting the Primitive Rational Points and Retaining Effective Equidistribution......Page 166
   10.4.4 Another Application of the Disjointness Result......Page 168
  References......Page 169
   11.1.1 Möbius Disjointness......Page 171
   11.1.2 Ergodic Theory Viewpoint on Sarnak's Conjecture......Page 173
   11.1.4 Sarnak's Conjecture a.e.......Page 175
  11.2 From a PNT in Dynamics to Sarnak's Conjecture......Page 176
   11.3.2 Dirichlet Convolution, Euler's Product......Page 180
   11.3.3 Distance Between Multiplicative Functions......Page 182
   11.3.4 Mean of a Multiplicative Function: The Prime Number Theorem (PNT)......Page 183
   11.3.5 Aperiodic Multiplicative Functions......Page 185
   11.3.6 Davenport Type Estimates on Short Intervals......Page 186
   11.3.7 The KBSZ Criterion......Page 187
   11.4.1 Formulation and Ergodic Interpretation......Page 189
   11.4.2 The Chowla Conjecture Implies Sarnak's Conjecture......Page 192
   11.4.3 The Logarithmic Versions of Chowla and Sarnak's Conjectures......Page 195
    11.4.4.1 Ergodicity of Measures for Which μ Is Quasi-Generic......Page 198
    11.4.4.2 Frantzikinakis' Results......Page 201
   11.4.5 Dynamical Properties of Furstenberg Systems Associated to the Liouville and Möbius Functions......Page 203
   11.5.1 The MOMO Property and Its Consequences......Page 205
   11.5.2 Möbius Disjointness and Entropy......Page 208
   11.5.3 The AOP Property and Its Consequences......Page 210
  11.6 Glimpses of Results on Sarnak's Conjecture......Page 211
    11.6.1.1 Horocycle Flows......Page 212
    11.6.1.2 Nilrotations, Affine Automorphisms......Page 214
    11.6.1.3 Other Algebraic Systems......Page 215
    11.6.2.2 Adic Systems and Bourgain's Criterion......Page 216
    11.6.2.3 Substitutions......Page 219
    11.6.2.4 Interval Exchanges......Page 220
    11.6.2.5 Systems of Rank One......Page 221
    11.6.2.6 Rokhlin Extensions......Page 223
   11.6.3 Distal Systems......Page 224
   11.6.4 Sub-polynomial Complexity......Page 226
   11.6.5 Systems of Number-Theoretic Origin......Page 227
    11.7.1.1 Sets of Multiples......Page 228
    11.7.1.3 Dynamics Comes Into Play......Page 229
    11.7.1.4 Further Generalizations......Page 230
    11.7.2.1 Mirsky Measure......Page 231
    11.7.2.4 All Invariant Measures......Page 232
   11.7.3 Topological Results......Page 233
    11.7.4.1 Polynomial Recurrence and Divisibility......Page 234
  References......Page 236
  12.1 Introduction......Page 244
   12.2.1 Ergodic Components of Members of V(x) and Vlog(x)......Page 245
   12.2.3 Ergodic Components of Measures in Vlog(x)......Page 248
  12.3 Relations Between Sarnak's Conjecture and the Chowla Conjecture......Page 249
  12.4 Examples......Page 250
  References......Page 254
  13.1 Introduction......Page 255
  13.2 Preliminaries......Page 256
  13.3 The Davenport-Erdös Theorem for Number Fields......Page 259
  References......Page 264
   14.1.1 Rational Approximations of Real Numbers......Page 265
   14.1.2 Integral Values of Binary Quadratic Forms......Page 268
   14.1.3 Best Rational Approximations and Continued Fractions......Page 269
   14.1.4 Perron's Characterization of Lagrange and Markov Spectra......Page 272
   14.1.5 Digression: Lagrange Spectrum and Cusp Excursions on the Modular Surface......Page 275
   14.1.6 Hall's Ray and Freiman's Constant......Page 278
   14.1.7 Statement of Moreira's Theorem......Page 279
   14.1.8 Hausdorff Dimension......Page 280
   14.2.1 Strategy of Proof of Moreira's Theorem......Page 281
   14.2.2 Dynamical Cantor Sets......Page 282
   14.2.3 Gauss-Cantor Sets......Page 283
   14.2.4 Non-essentially Affine Cantor Sets......Page 284
   14.2.5 Moreira's Dimension Formula......Page 286
   14.2.6 First Step Towards Moreira's Theorem 14.36: Projections of Gauss-Cantor Sets......Page 287
   14.2.7 Second Step Towards Moreira's Theorem 14.36: Upper Semi-continuity......Page 289
   14.2.8 Third Step Towards Moreira's Theorem 14.36: Lower Semi-continuity......Page 290
   14.2.9 End of Proof of Moreira's Theorem 14.36......Page 293
  Appendix 1: Proof of Hurwitz Theorem......Page 294
  References......Page 296
  15.1 Introduction......Page 298
  15.2 Meissel and Gram......Page 299
  15.3 Generalizing Meissel's Proof, I......Page 300
   15.3.1 An Intriguing Example......Page 302
   15.4.1 A Related Problem......Page 306
   15.4.2 The Horizon......Page 309
  15.6 Generalizing Meissel's Proof, II......Page 310
   15.6.1 The Problem at Large......Page 311
   15.6.2 Beginning of a Theory When F=1......Page 312
   15.6.3 The Localization Problem, Case F=1......Page 313
   15.6.4 From Λ to μ/From ψ to M......Page 314
  15.7 Generalizing Meissel's Proof, III......Page 315
   15.7.1 A Related Problem......Page 317
  References......Page 318
  16.1 Historical Setting......Page 321
  16.2 Lemmas......Page 323
  16.3 Proof of the Theorem 16.8......Page 326
  References......Page 327
Part III  Selected Topics in Dynamics......Page 328
  17.1 Historical Perspective......Page 329
  17.2 Preliminary Definitions......Page 330
  17.3 Mixing Properties for Probability-Preserving Transformations......Page 332
   17.3.1 Mixing Property Potpourri......Page 333
   17.3.2 Weak Mixing Equivalences......Page 334
  17.4 The Terrain of Infinite Rank-One......Page 335
   17.5.1 Weakly Wandering Sets and Obstacles to Strong Mixing......Page 337
    17.6.1.1 Weak Mixing and Spectral Weak Mixing......Page 339
    17.6.1.2 Weakly Doubly Ergodic......Page 340
    17.6.2.2 Infinite Ergodic Index......Page 341
    17.6.3.1 Koopman Mixing......Page 342
    17.6.3.2 K-automorphisms......Page 343
    17.6.3.3 Rational Weak Mixing......Page 344
    17.6.3.4 Multiple Recurrence......Page 345
    17.6.3.5 Measurable Sensitivity......Page 346
  References......Page 347
  18.1 Introduction......Page 352
  18.2 Preliminaries......Page 354
   18.2.1 Background on Fragmentability and Tame Families......Page 355
   18.2.2 Independent Sequences of Functions......Page 356
   18.2.3 More Properties of Fragmented Families......Page 358
  18.3 Classes of Dynamical Systems......Page 360
   18.3.1 Some Classes of Functions......Page 361
   18.3.2 Cyclic S-compactifications......Page 362
   18.4.1 Symbolic Systems and Coding Functions......Page 363
   18.4.2 Tame and HNS Subsets of Z......Page 367
  18.5 Entropy and Null Systems......Page 368
  18.6 Some Examples of Tame Functions and Systems......Page 370
   18.6.1 A Special Class of Generalized Sturmian Systems......Page 377
   18.6.2 Strong Almost 1-1 Equivalence and Tameness......Page 380
   18.7.1 Order Preserving Action on the Unit Interval......Page 382
   18.7.2 Circularly Ordered Systems......Page 383
   18.7.3 Noncommutative Sturmian Like Symbolic Systems......Page 384
  18.8 Tame Minimal Systems and Topological Groups......Page 385
  References......Page 390
  19.1 Introduction......Page 394
  19.2 Piecewise Rotation of the Circle......Page 398
  19.3 IPR Map......Page 408
  19.4 p and σπ via {ζj, ζj*} and {ξj, ξj*}......Page 420
  19.5 Castles and Zippered Rectangles......Page 424
  References......Page 432