6 1. Higher order Fourier analysis

Corollary 1.1.3. Let x: N →

Td.

Then x is asymptotically equidistributed

in

Td

if and only if, for each k ∈

Zd\{0},

the sequence n → k · x(n) is

asymptotically equidistributed in T.

Exercise 1.1.4. Show that a sequence x : N →

Td

is totally asymptotically

equidistributed if and only if one has

(1.3) lim

N→∞

En∈[N]e(k · x(n))e(αn) = 0

for all k ∈

Zd\{0}

and all rational α.

This quickly gives a test for equidistribution for linear sequences, some-

times known as the equidistribution theorem:

Exercise 1.1.5. Let α, β ∈

Td.

By using the geometric series formula, show

that the following are equivalent:

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

(ii) The sequence n → nα + β is totally asymptotically equidistributed

on N.

(iii) The sequence n → nα + β is totally asymptotically equidistributed

on Z.

(iv) α is irrational, in the sense that k · α = 0 for any non-zero k ∈

Zd.

Remark 1.1.4. One can view Exercise 1.1.5 as an assertion that a linear

sequence xn will equidistribute itself unless there is an “obvious” algebraic

obstruction to it doing so, such as k · xn being constant for some non-zero

k. This theme of algebraic obstructions being the “only” obstructions to

uniform distribution will be present throughout the text.

Exercise 1.1.5 shows that linear sequences with irrational shift α are

equidistributed. At the other extreme, if α is rational in the sense that

mα = 0 for some positive integer m, then the sequence n → nα + β is

clearly periodic of period m, and definitely not equidistributed.

In the one-dimensional case d = 1, these are the only two possibili-

ties. But in higher dimensions, one can have a mixture of the two ex-

tremes, that exhibits irrational behaviour in some directions and periodic

behaviour in others. Consider for instance the two-dimensional sequence

n → (

√

2n,

1

2

n) mod

Z2.

The first coordinate is totally asymptotically

equidistributed in T, while the second coordinate is periodic; the shift

(

√

2,

1

2

) is neither irrational nor rational, but is a mixture of both. As such,

we see that the two-dimensional sequence is equidistributed with respect to

Haar measure on the group T × (

1

2

Z/Z).

This phenomenon generalises: