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