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 dμ

∞

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: