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-
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
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
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.
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