Number Theory: An Introduction to Mathematics

472 XI Uniform Distribution and Ergodic Theory

4 Applications................................................

We now give some examples to illustrate the general concepts and results of the

previous section.

(i) SupposeX=Rd/Zdis ad-dimensional torus,Bis the family ofBorel subsetsof

X(i.e., theσ-algebra of subsets generated by the family of open sets), andμ=λis

Lebesgue measure, i.e.μ(B)=

∫

XχB(x)dxfor anyB∈B,whereχBis the indica-

tor function ofB.Everyx∈Xis represented by a unique vector(ξ 1 ,...,ξd),where

0 ≤ξk< 1 (k= 1 ,...,d),andXis an abelian group with additionz=x+ydefined

byζk≡ξk+ηkmod 1(k= 1 ,...,d).

For anya∈X,thetranslation Ta:X→Xdefined byTax=x+ais a measure-

preserving transformation of the probability space(X,B,λ).

Proposition 19The translation Ta:X→X of the d-dimensional torus X=Rd/Zd

is ergodic if and only if 1 ,α 1 ,...,αdare linearly independent over the rational field

Q,where(α 1 ,...,αd)is the vector which represents a.

Proof Suppose first that 1,α 1 ,...,αdare not linearly independent overQ. Then there

exists a nonzero vectorn∈Zdsuch that

n·a=v 1 α 1 +···+vdαd∈Z.

Hence iff(x)=e(n·x),thenf(Tax)=f(x)for allx.Sincefis not constant a.e.,

it follows from part (i) of Proposition 18 thatTais not ergodic.

Suppose on the other hand that 1,α 1 ,...,αdare linearly independent overQand

letfbe an integrable function such thatf(Tax)= f(x)a.e. Thenf(Tax)and f(x)

have the same Fourier coefficients:

∫

X

f(x)e(−n·x)dx=

∫

X

f(x+a)e(−n·x)dx=e(n·a)

∫

X

f(x)e(−n·x)dx.

Sincee(n·a)=1foralln=0, it follows that

∫

X

f(x)e(−n·x)dx=0foralln= 0.

Since integrable functions with the same Fourier coefficients must agree almost

everywhere, this proves thatf is constant a.e. Hence, by Proposition 18 again,Ta

is ergodic. 

If we compare Proposition 3′and the remarks after its proof with Proposition 19,
then we see from Theorems 1′-2′and Proposition 18 that the following five statements
are equivalent forX=Rd/Zdand anya∈X:
(α) the sequence({na})is dense inX;
(β)foreveryx∈X, the sequence(x+na)is uniformly distributed inX;
(γ) the translationTa:X→Xis ergodic;
(δ) for each continuous functionf:X→C,limn→∞n−^1

∑n− 1

k= 0 f(T

k

ax)=

∫

Xfdλ

for allx∈X;