1.1. Equidistribution in tori 5
Exercise 1.1.3. Let x: N X be a sequence into a compact metric space
X which is equidistributed relative to some probability measure μ. Let
ϕ: R R be a compactly supported, piecewise continuous function with
only finitely many pieces. Show that for any f C(X) one has
lim
N→∞
1
N
n∈N
ϕ(n/N)f(x(n)) =
X
f

0
ϕ(t) dt
and for any open U whose boundary has measure zero, one has
lim
N→∞
1
N
n∈N:x(n)∈U
ϕ(n/N) = μ(U)

0
ϕ(t) dt .
In this section, X will be a torus (i.e., a compact connected abelian Lie
group), which from the theory of Lie groups is isomorphic to the standard
torus
Td,
where d is the dimension of the torus. This torus is then equipped
with Haar measure, which is the unique Borel probability measure on the
torus which is translation-invariant. One can identify the standard torus
Td
with the standard fundamental domain [0,
1)d,
in which case the Haar mea-
sure is equated with the usual Lebesgue measure. We shall call a sequence
x1,x2,... in
Td
(asymptotically) equidistributed if it is (asymptotically)
equidistributed with respect to Haar measure.
We have a simple criterion for when a sequence is asymptotically equidis-
tributed, that reduces the problem to that of estimating exponential sums:
Proposition 1.1.2 (Weyl equidistribution criterion). Let x: N
Td.
Then x is asymptotically equidistributed if and only if
(1.2) lim
N→∞
En∈[N]e(k · x(n)) = 0
for all k
Zd\{0},
where e(y) :=
e2πiy.
Here we use the dot product
(k1,...,kd) · (x1,...,xd) := k1x1 + · · · + kdxd
which maps
Zd
×
Td
to T.
Proof. The “only if” part is immediate from (1.1). For the “if” part, we
see from (1.2) that (1.1) holds whenever f is a plane wave f(y) := e(k · y)
for some k
Zd
(checking the k = 0 case separately), and thus by linearity
whenever f is a trigonometric polynomial. But by Fourier analysis (or from
the Stone-Weierstrass theorem), the trigonometric polynomials are dense
in
C(Td)
in the uniform topology. The claim now follows from a standard
limiting argument.
As one consequence of this proposition, one can reduce multidimensional
equidistribution to single-dimensional equidistribution:
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