10 1. Higher order Fourier analysis
We discussed recurrence for one-dimensional sequences x: n x(n).
It is also of interest to establish an analogous theory for multi-dimensional
sequences, as follows.
Definition 1.1.10. A multidimensional sequence x:
Zm
X is asymptot-
ically equidistributed relative to a probability measure μ if, for every con-
tinuous, compactly supported function ϕ:
Rm
R and every function
f C(X), one has
1
N m
n∈Zm
ϕ(n/N)f(x(n))
Rm
ϕ
X
f
as N ∞. The sequence is totally asymptotically equidistributed relative
to μ if the sequence n x(qn+r) is asymptotically equidistributed relative
to μ for all positive integers q and all r
Zm.
Exercise 1.1.9. Show that this definition of equidistribution on
Zm
co-
incides with the preceding definition of equidistribution on Z in the one-
dimensional case m = 1.
Exercise 1.1.10 (Multidimensional Weyl equidistribution criterion). Let
x:
Zm

Td
be a multidimensional sequence. Show that x is asymptotically
equidistributed if and only if
(1.4) lim
N→∞
1
N m
n∈Zm:n/N∈B
e(k · x(n)) = 0
for all k
Zd\{0}
and all rectangular boxes B in
Rm.
Then show that x is
totally asymptotically equidistributed if and only if
(1.5) lim
N→∞
1
N m
n∈Zm:n/N∈B
e(k · x(n))e(α · n) = 0
for all k
Zd\{0},
all rectangular boxes B in
Rm,
and all rational α
Qm.
Exercise 1.1.11. Let α1,...,αm,β
Td,
and let x:
Zm

Td
be the linear
sequence x(n1,...,nm) := n1α1 + · · · + nmαm + β. Show that the following
are equivalent:
(i) The sequence x is asymptotically equidistributed on
Zm.
(ii) The sequence x is totally asymptotically equidistributed on
Zm.
(iii) We have (k · α1,...,k · αm) = 0 for any non-zero k
Zd.
Exercise 1.1.12 (Multidimensional van der Corput lemma). Let x:
Zm

Td
be such that the sequence ∂hx: n x(n + h) x(n) is asymptotically
equidistributed on
Zm
for all h outside of a hyperplane in
Rm.
Show that
x is asymptotically equidistributed on
Zm.
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