6 1. Higher order Fourier analysis
Corollary 1.1.3. Let x: N →
Then x is asymptotically equidistributed
if and only if, for each k ∈
the sequence n → k · x(n) is
asymptotically equidistributed in T.
Exercise 1.1.4. Show that a sequence x : N →
is totally asymptotically
equidistributed if and only if one has
En∈[N]e(k · x(n))e(αn) = 0
for all k ∈
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 ∈
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
(iii) The sequence n → nα + β is totally asymptotically equidistributed
(iv) α is irrational, in the sense that k · α = 0 for any non-zero k ∈
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 → (
The first coordinate is totally asymptotically
equidistributed in T, while the second coordinate is periodic; the shift
) 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 × (
This phenomenon generalises: