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 dμ

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.