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 nα

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