22 1. Higher order Fourier analysis

for all standard continuous functions f ∈ C(X). Here, we define the expec-

tation of a limit function in the obvious limit manner, thus

En∈[N]f(x(n)) = lim

α→α∞

En∈[Nα]f(xα(n))

if N = limα→α∞ Nα and x = limα→α∞ xα.

We say that x is totally equidistributed relative to μ if the sequence

n → x(qn + r) is equidistributed on [N/q] for every standard q 0 and

r ∈ Z (extending x arbitrarily outside [N] if necessary).

Remark 1.1.24. One could just as easily replace the space of continuous

functions by any dense subclass in the uniform topology, such as the space

of Lipschitz functions.

The ultralimit notion of equidistribution is closely related to that of

both asymptotic equidistribution and single-scale equidistribution, as the

following exercises indicate:

Exercise 1.1.28 (Asymptotic equidistribution vs. ultralimit equidistribu-

tion). Let x: N → X be a sequence into a standard compact metric space

(which can then be extended from a map from

∗N

to

∗X

as usual), let μ be

a Borel probability measure on X. Show that x is asymptotically equidis-

tributed on N with respect to μ if and only if x is equidistributed on [N]

for every unbounded natural number N and every choice of non-principal

ultrafilter α∞.

Exercise 1.1.29 (Single-scale equidistribution vs. ultralimit equidistribu-

tion). For every α ∈ N, let Nα be a natural number that goes to infinity as

α → ∞, let xα : [Nα] → X be a map to a standard compact metric space.

Let μ be a Borel probability measure on X. Write N := limα→α∞ Nα and

x := limα→α∞ xα for the ultralimits. Show that x is equidistributed with

respect to μ if and only if, for every standard δ 0, xα is δ-equidistributed

with respect to μ for all α suﬃciently close to α∞.

In view of these correspondences, it is thus not surprising that one has

ultralimit analogues of the asymptotic and single-scale theory. These ana-

logues tend to be logically equivalent to the single-scale counterparts (once

one concedes all quantitative bounds), but are formally similar (though not

identical) to the asymptotic counterparts, thus providing a bridge between

the two theories, which we can summarise by the following three statements:

(i) Asymptotic theory is analogous to ultralimit theory (in particular,

the statements and proofs are formally similar);

(ii) ultralimit theory is logically equivalent to qualitative finitary the-

ory; and