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