24 1. Higher order Fourier analysis
x(n + h) x(n) is equidistributed on [N] for H values of h [H] (by
extending x arbitrarily outside of [N]). Show that x is equidistributed on
[N]. Similarly, “equidistributed” is replaced by “totally equidistributed”.
Here is the analogue of the Vinogradov lemma:
Exercise 1.1.36 (Vinogradov lemma, ultralimit version). Let α ∗T, N
be unbounded, and ε 0 be infinitesimal. Suppose that
T
ε for
N values of n [−N, N]. Show that there exists a positive standard
integer q such that αq
T
ε/N.
These two lemmas allow us to establish the ultralimit polynomial equidis-
tribution theory:
Exercise 1.1.37. Let P : Z
∗Td
be a polynomial sequence P (n) :=
αsns
+ · · · + α0 with s, d standard, and α0,...,αs
∗Td.
Let N be an
unbounded natural number. Suppose that P is not totally equidistributed
on [N]. Show that there exists a non-zero standard k
Zd
with k ·αs
T
N
−s.
Exercise 1.1.38. With the hypotheses of Exercise 1.1.37, show in fact that
there exists a non-zero standard k
Zd
such that k · αi
T
N
−i
for all
i = 0,...,s.
Exercise 1.1.39 (Ultralimit equidistribution theorem for abelian polyno-
mial sequences). Let P be a polynomial map from ∗Z to
∗Td
of some stan-
dard degree s 0. Let N be an unbounded natural number. Then there
exists a decomposition
P = Psmth + Pequi + Prat
into polynomials of degree s, where
(i) (Psmth is smooth) The
ith
coefficient αi,smth of Psmth has size
O(N
−i).
In particular, on the interval [N], Psmth is Lipschitz with
homogeneous norm O(1/N).
(ii) (Pequi is equidistributed) There exists a standard subtorus T of
Td,
such that Pequi takes values in T and is totally equidistributed on
[N] in this torus.
(iii) (Prat is rational) The coefficients αi,rat of Prat are standard rational
elements of
Td.
In particular, there is a standard positive integer
q such that qPrat = 0 and Prat is periodic with period q.
Exercise 1.1.40. Show that the torus T is uniquely determined by P , and
decomposition P = Psmth + Pequi + Prat in Exercise 1.1.39 is unique up to
expressions taking values in T (i.e., if one is given another decomposition
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