1.6. Inverse conjecture over the integers 93

a polynomial gΓ: H → G/Γ into a nilmanifold G/Γ, and then composing

this with a

Lipschitz18

function F : G/Γ → C. These classes of sequences

certainly include the polynomial phases, but are somewhat more general;

for instance, they

almost19

include bracket polynomial phases such as n →

e( αn βn).

In this section we set out the basic theory for these nilsequences, in-

cluding their equidistribution theory (which generalises the equidistribution

theory of polynomial flows on tori from Section 1.1) and show that they

are indeed obstructions to the Gowers norm being small. This leads to the

inverse conjecture for the Gowers norms that shows that the Gowers norms

on cyclic groups are indeed controlled by these sequences.

1.6.1. General theory of polynomial maps. In previous sections, we

defined the notion of a (non-classical) polynomial map φ of degree at most

d between two additive groups H, G, to be a map φ: H → G obeying the

identity

∂h1 . . . ∂hd+1 φ(x) = 0

for all x, h1,...,hd+1 ∈ H, where ∂hφ(x) := φ(x + h) − φ(x) is the additive

discrete derivative operator.

There is another way to view this concept. For any k, d ≥ 0, define

the Host-Kra group

HKk(H,

≤ d) of H of dimension k and degree d to be

the subgroup of

H{0,1}d

consisting of all tuples (xω)ω∈{0,1}k obeying the

constraints

ω∈F

(−1)|ω|xω

= 0

for all faces F of the unit cube {0, 1}k of dimension at least d + 1, where

|(ω1,...,ωk)| := ω1 + · · · + ωk. (These constraints are of course trivial if

k ≤ d.) An r-dimensional face of the unit cube {0,

1}k

is of course formed

by freezing k −r of the coordinates to a fixed value in {0, 1}, and letting the

remaining r coordinates vary freely in {0, 1}.

Thus, for instance,

HK2(H,

≤ 1) is (essentially) the space of parallelo-

grams (x, x + h, x + k, x + h + k) in

H4,

while

HK2(H,

≤ 0) is the diagonal

group {(x, x, x, x) : x ∈

H4},

and

HK2(H,

≤ 2) is all of

H4.

Exercise 1.6.1. Let φ: H → G be a map between additive groups, and let

k d ≥ 0. Show that φ is a (non-classical) polynomial of degree at most d if

it maps

HKk(H,

≤ 1) to

HKk(G,

≤ d), i.e., that (φ(xω))ω∈{0,1}k ∈

HKk(G,

≤

d) whenever (xω)ω∈{0,1}k ∈

HKk(H,

≤ 1).

18The

Lipschitz regularity class is convenient for minor technical reasons, but one could also

use other regularity classes here if desired.

19The

“almost” here is because the relevant functions F : G/Γ → C involved are only piece-

wise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view

bracket polynomial phases as “morally” being nilsequences.