Index

abstract measure-preserving system, 177

ad` eles, 29

almost periodic, 90

almost periodic function, 172

almost prime, 245, 269

amalgamated free product, 43

amenable group, 3

Archimedean, 23

baby Furstenberg structure theorem, 135

baker’s map, 77

Baker-Campbell-Hausdorff formula, 221

Bernoulli system, 76

Birkhoff ergodic theorem, 154

Birkhoff recurrence theorem, 79, 91

Bombieri-Vinogradov theorem, 264

Borel probability measure, 140

Borel-Cantelli lemma, 17

bracket polynomial, 257

Cantor space, 27

Cayley graph, 3, 271

Ces` aro convergence, 182

chain, 69

chain complex, 69

Chebyshev’s inequality, 16

Chen’s theorem, 246

coboundary, 68

cochain, 70

cocycle, 68, 126

color focusing, 101

commutator, 217

compact extension, 200

compact system, 174

conditional expectation, 149

conditional weak mixing, 206

confidence level, 47

Cram´ er conjecture, 260

Cram´ er’s random model, 241

curse of dimensionality, 143

cycle, 69

density Hales-Jewett theorem, 171

density Ramsey theorems, 80

dichotomy between structure and

randomness, 138, 190, 209, 227

discrete logarithm, 38

disintegration, 161

distal measure-preserving system, 213

distal system, 134

divisor bound, 32

dual function, 150, 210

Dunford-Schwartz maximal inequality, 152

dyadic interval, 57

dyadic pigeonhole principle, 19

dynamical system, 75

elementary subgroup, 269

Elliott-Halberstam conjecture, 264

Ellis-Nakamura lemma, 113

equicontinuous system, 119

equidistribution, 9

ergodic, 156

decomposition, 162

theory, 78

Euler product formula, 240

Euler totient function, 37

expander graph, 271

Fermat prime, 38

Fermat’s little theorem, 37

first moment method, 16

291