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)
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