104 1. Higher order Fourier analysis

of degree ≤ s with a vertical frequency of 0. Finally, observe that the

space of degree ≤ s nilsequences with a vertical frequency is closed under

multiplication and complex conjugation.

Exercise 1.6.19. Show that a degree ≤ 1 nilsequence with a vertical fre-

quency necessarily takes the form ψ(n) = ce(αn) for some c ∈ C and α ∈ R

(and conversely, all such sequences are degree ≤ 1 nilsequences with a ver-

tical frequency). Thus, up to constants, degree ≤ 1 nilsequences with a

vertical frequency are the same as Fourier characters.

A basic fact (generalising the invertibility of the Fourier transform in the

degree ≤ 1 case) is that the nilsequences with vertical frequency generate

all the other nilsequences:

Exercise 1.6.20. Show that any degree ≤ s nilsequence can be approxi-

mated to arbitrary accuracy in the uniform norm by a linear combination

of nilsequences with a vertical frequency. (Hint: Use the Stone-Weierstrass

theorem.)

More quantitatively, show that a degree ≤ s nilsequence of complex-

ity ≤ M can be approximated uniformly to error ε by a sum of OM,ε,s(1)

nilsequences, each with a representation with a vertical frequency that is

of complexity OM,ε,s(1). (Hint: This can be deduced from the qualitative

result by a compactness argument using the Arzel´ a-Ascoli theorem.)

A derivative Δhe(P (n)) of a polynomial phase is a polynomial phase of

one lower degree. There is an analogous fact for nilsequences with a vertical

frequency:

Lemma 1.6.13 (Differentiating nilsequences with a vertical frequency). Let

s ≥ 1, and let ψ be a degree ≤ s nilsequence with a vertical frequency. Then

for any h ∈ Z, Δhψ is a degree ≤ s − 1 nilsequence. Furthermore, if ψ has

complexity ≤ M (with a vertical frequency representation), then Δhψ has

complexity OM,s(1).

Proof. We just prove the first claim, as the second claim follows by refining

the argument.

We write ψ = F (g(n)Γ) for some polynomial sequence g : Z → G/Γ and

some Lipschitz function F with a vertical frequency. We then express

Δhψ(n) =

˜(˜(n)(Γ

F g × Γ))

where

˜

F : G × G/(Γ × Γ) → C is the function

˜(x,

F y) := F (x)F (y)