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 + β is asymptotically equidistributed on N.
(ii) The sequence n + β is totally asymptotically equidistributed
on N.
(iii) The sequence n + 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
= 0 for some positive integer m, then the sequence n + 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:
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