108 1. Higher order Fourier analysis
Proof. It suffices to show that, for each Lipschitz function F : G/Γ C,
that
lim
n→∞
En∈[N]F (x(n)) =
G/Γ
F dμG/Γ.
By Exercise 1.6.20, we may assume that F has a vertical frequency. If this
vertical frequency is non-zero, then F descends to a function on the degree
s 1 filtered nilmanifold G/GsΓ, and the claim then follows from the
equidistribution hypothesis on this space. So suppose instead that F has
a non-zero vertical frequency. By vertically rotating F (and using the Gs-
invariance of μG/Γ we conclude that
G/Γ
FμG/Γ = 0. Applying the van der
Corput inequality (Lemma 1.1.6), we now see that it suffices to show that
lim
n→∞
En∈[N]F (x(n + h))F (x(n)) = 0
for each non-zero h. The function (x, y) F (x)F (y) on G/Γ × G/Γ is
T
Δ-invariant
(because of the vertical frequency hypothesis) and so descends
to a function
˜
F on (G/Γ × G/Γ)/T
Δ.
We thus have
lim
n→∞
En∈[N]F (x(n + h))F (x(n)) =
(G/Γ×G/Γ)/T
Δ
˜
F dμh.
The function
˜
F has a non-zero vertical frequency with respect to the residual
action of T (or more precisely, of (T ×
T)/TΔ,
which is isomorphic to T).
As μh is invariant with respect to this action, the integral thus vanishes, as
required.
This gives a useful criterion for equidistribution of polynomial orbits.
Define a horizontal character to be a continuous homomorphism η from G
to R/Z that annihilates Γ (or equivalently, an element of the Pontryagin
dual of the horizontal torus G/([G, G]Γ)). This is easily seen to be a torus.
Let πi : G≥i Ti be the projection map.
Theorem 1.6.16 (Leibman equidistribution criterion). Let O : n g(n)Γ
be a polynomial orbit on a degree s filtered nilmanifold G/Γ. Suppose that
G = G≥0 = G≥1. Then O is asymptotically equidistributed in G/Γ if and
only if η g is non-constant for each non-trivial horizontal character.
This theorem was first established by Leibman [Le2005] (by a slightly
different method), and also follows from the above van der Corput lemma
and some tedious additional computations; see [GrTa2011] for details. For
linear orbits, this result was established in [Pa1970], [Gr1961]. Using this
criterion (together with more quantitative analogues for single-scale equidis-
tribution), one can develop equidistribution decompositions that generalise
those in Section 1.1. Again, the details are technical and we will refer to
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