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
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