1.1. Equidistribution in tori 23

(iii) quantitative finitary theory is a strengthening of qualitative finitary

theory.

For instance, here is the ultralimit version of the Weyl criterion:

Exercise 1.1.30 (Ultralimit Weyl equidistribution criterion). Let x: [N] →

∗Td

be a limit function for some unbounded N and standard d. Then x is

equidistributed if and only if

(1.7) En∈[N]e(k · x(n)) = o(1)

for all standard k ∈

Zd\{0}.

Hint: Mimic the proof of Proposition 1.1.2.

Exercise 1.1.31. Use Exercise 1.1.30 to recover a weak version of Propo-

sition 1.1.13, in which the quantities

δcd

,

δCd

are replaced by (ineffective)

functions of δ that decay to zero as δ → 0. Conversely, use this weak version

to recover Exercise 1.1.30. (Hint: Similar arguments appear in Section 2.1.)

Exercise 1.1.32. With the notation of Exercise 1.1.30, show that x is to-

tally equidistributed if and only if

En∈[N]e(k · x(n))e(θn) = o(1)

for all standard k ∈

Zd\{0}

and standard rational θ.

Exercise 1.1.33. With the notation of Exercise 1.1.30, show that x is

equidistributed in

Td

on [N] if and only if k · x is equidistributed in T

on [N] for every non-zero standard k ∈

Zd.

Now we establish the ultralimit version of the linear equidistribution

criterion:

Exercise 1.1.34. Let α, β ∈

∗Td,

and let N be an unbounded integer.

Show that the following are equivalent:

(i) The sequence n → nα + β is equidistributed on [N].

(ii) The sequence n → nα + β is totally equidistributed on [N].

(iii) α is irrational to scale 1/N, in the sense that k · α = O(1/N) for

any non-zero standard k ∈

Zd.

Note that in the ultralimit setting, assertions such as k · α = O(1/N)

make perfectly rigorous sense (it means that |k·α| ≥ C/N for every standard

C), but when using finitary asymptotic big-O notation

Next, we establish the analogue of the van der Corput lemma:

Exercise 1.1.35 (van der Corput lemma, ultralimit version). Let N be an

unbounded integer, and let x: [N] →

∗Td

be a limit sequence. Let H =

o(N) be unbounded, and suppose that the derivative sequence ∂hx: n →