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

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.
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