186 Index

equidistribution (ultralimit analysis), 22

equidistribution theorem, 6

error correction of polynomials, 67

exponential sum, 2

factor, 33

Fej´ er summation, 13

filtered group, 94

filtration, 94

Fourier measurability, 35

Fourier pseudorandomness, 48

fragmentation, 31

generalised von Neumann inequality, 59

generalised von Neumann theorem, 118

Gowers box space, 154

Gowers inner product, 57, 154

Gowers triangle inequality, 157

Gowers uniformity norm, 47, 57, 154

Gowers’ Cauchy-Schwarz argument, 83

Gowers-Host-Kra semi-norm, 155

Gowers-Wolf theorem, 92

Gromov’s theorem, 145

growth spurt, 19

Haar measure, 5

Hall-Petresco formula, 98

Hamilton’s equation of motion, 176

Hardy-Littlewood maximal inequality,

36

Heisenberg group, 98

Heisenberg nilmanifold, 100

Higher order inner product space, 153

Hilbert cube lemma, 46

Hilbert space, 150

horizontal character, 108

horizontal torus, 108

Host-Kra group, 93, 94

Host-Kra measure, 155

hyperreal, 135

indicator function, 2

infinitesimal, 21, 138

inner product space, 149

inverse conjecture for the Gowers norm,

58, 106

inverse conjecture for the Gowers

uniformity norms, 75

irrational, 6

join, 34

Kronecker factor, 42

Kronecker measurability, 42

Lazard-Leibman theorem, 96

Liebman equidistribution criterion, 108

limit finite set, 20

limit function, 21

limit number, 20

limit object, 134

limit set, 20

linear phase, 2

linear forms conditions, 118

Lipschitz norm, 11

Littlewood-Paley theory, 171

local Gowers inner product, 160

local testability, 79

Loeb measure, 41

Los’s theorem, 20

low rank, 65

lower central series, 94

lower face, 95

Lucas’ theorem, 80

Mal’cev basis, 99

multiple recurrence, 19

Newton’s second law, 176

nilpotent, 98

Noetherian condition, 142

non-principal ultrafilter, 133

norm, 150

phase heuristic, 167

polyar body, 168

polynomial, 54, 60

polynomial orbit, 101

polynomial phase invariance, 59

polynomial recurrence, 9

polynomial sequence, 98

Pontryagin dual, 57

Ramsey’s theorem, 80

recurrence, 19

refinement, 34

regularity lemma for polynomials, 71

relative van der Corput lemma, 107

reproducing formula, 170

restriction estimate, 114

rigidity, 67

Roth’s theorem, 26, 33, 112

Roth’s theorem in the primes, 110

Roth-pseudorandom, 112

Schr¨ odinger equation, 172, 175