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.
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