102 1. Higher order Fourier analysis

(using the notation from Exercise 1.6.15) is a polynomial sequence in the

Heisenberg nilmanifold.

With the above example, we see the emergence of bracket polynomials

when representing polynomial orbits in a fundamental domain. Indeed, one

can view the entire machinery of orbits in nilmanifolds as a means of eﬃ-

ciently capturing such polynomials in an algebraically tractable framework

(namely, that of polynomial sequences in nilpotent groups). The piecewise

continuous nature of the bracket polynomials is then ultimately tied to the

twisted gluing needed to identify the fundamental domain with the nilman-

ifold.

Finally, we can define the notion of a (basic Lipschitz) nilsequence of

degree ≤ s. This is a sequence ψ : Z → C of the form ψ(n) := F (O(n)),

where O : Z → G/Γ is a polynomial orbit in a filtered nilmanifold of degree

≤ s, and F : G/Γ → C is a

Lipschitz21

function. We say that the nilsequence

has complexity at most M if the filtered nilmanifold has complexity at most

M, and the (inhomogeneous Lipschitz norm) of F is also at most M.

A basic example of a degree ≤ s nilsequence is a polynomial phase

n → e(P (n)), where P : Z → R/Z is a polynomial of degree ≤ s. A bit more

generally, n → F (P (n)) is a degree ≤ s sequence, whenever F : R/Z → C

is a Lipschitz function. In view of Exercises 1.6.15, 1.6.16, we also see that

(1.52) n → e(αn βn )ψ({αn})ψ({βn})

or, more generally,

n → F (αn βn )ψ({αn})ψ({βn})

are also degree ≤ 2 nilsequences, where ψ : [0, 1] → C is a Lipschitz function

that vanishes near 0 and 1. The ψ({αn}) factor is not needed (as there is

no twisting in the x coordinate in Exercise 1.6.15), but the ψ({βn}) factor

is (unfortunately) necessary, as otherwise one encounters the discontinuity

inherent in the βn term (and one would merely have a piecewise Lipschitz

nilsequence rather than a genuinely Lipschitz nilsequence). Because of this

discontinuity, bracket polynomial phases n → e(αn βn ) cannot quite be

viewed as Lipschitz nilsequences, but from a heuristic viewpoint it is often

helpful to pretend as if bracket polynomial phases are model instances of

nilsequences.

The only degree ≤ 0 nilsequences are the constants. The degree ≤ 1

nilsequences are essentially the quasiperiodic functions:

21One needs a metric on G/Γ to define the Lipschitz constant, but this can be done, for

instance, by using a basis e1, . . . , ed of Γ to identify G/Γ with a fundamental domain [0, 1]d, and

using this to construct some (artificial) metric on G/Γ. The details of such a construction will

not be important here.