1.6. Inverse conjecture over the integers 107

it will be of importance to be able to compute statistics of nilsequences ψ,

such as their averages En∈[N]ψ(n) for a large integer N; this generalises

the computation of exponential sums such as En∈[N]e(P (n)) that occurred

in Section 1.1. This is closely related to the equidistribution of polynomial

orbits O : Z → G/Γ in nilmanifolds. Note that as G/Γ is a compact quotient

of a locally compact group G, it comes endowed with a unique left-invariant

Haar measure μG/Γ (which is isomorphic to the Lebesgue measure on a

fundamental domain [0,

1]d

of that nilmanifold). By default, when we talk

about equidistribution in a nilmanifold, we mean with respect to the Haar

measure; thus O is asymptotically equidistributed if and only if

lim

N→∞

En∈[N]F (O(n)) = 0

for all Lipschitz F : G/Γ → C. One can also describe single-scale equidis-

tribution (and non-standard equidistribution) in a similar fashion, but for

the sake of discussion let us restrict our attention to the simpler and more

classical situation of asymptotic equidistribution here (although it is the

single-scale equidistribution theory which is ultimately relevant to questions

relating to the Gowers norms).

When studying equidistribution of polynomial sequences in a torus

Td,

a

key tool was the van der Corput lemma (Lemma 1.1.6). This lemma asserted

that if a sequence x: Z →

Td

is such that all derivatives ∂hx: Z →

Td

with

h = 0 are asymptotically equidistributed, then x itself is also asymptotically

equidistributed.

The notion of a derivative requires the ability to perform subtraction

on the range space

Td

: ∂hx(n + h) − ∂hx(n). When working in a higher

degree nilmanifold G/Γ, which is not a torus, we do not have a notion of

subtraction. However, such manifolds are still torus bundles with torus

T := G≥s/Γ≥s. This gives a weaker notion of subtraction, namely the

map π : G/Γ × G/Γ → (G/Γ ×

G/Γ)/TΔ,

where

TΔ

is the diagonal action

gs : (x, y) → (gsx, gsy) of the torus T on the product space G/Γ×G/Γ. This

leads to a generalisation of the van der Corput lemma:

Lemma 1.6.15 (Relative van der Corput lemma). Let x: Z → G/Γ be a

sequence in a degree ≤ s nilmanifold for some s ≥ 1. Suppose that the

projection of x to the degree ≤ s − 1 filtered nilmanifold G/GsΓ is asymp-

totically equidistributed, and suppose also that for each non-zero h ∈ Z, the

sequence ∂hx: n → π(x(n + h),x(n)) is asymptotically equidistributed with

respect to some T-invariant measure μh on (G/Γ ×

G/Γ)/TΔ.

Then x is

asymptotically equidistributed in G/Γ.