Number Theory: An Introduction to Mathematics

4 Applications 473

(ε) for each function f ∈ L(X,B,λ), limn→∞n−^1

∑n− 1

k= 0 f(T

k

ax)=

∫

Xfdλfor

almost allx∈X.

(ii) Again supposeX =Rd/Zdis ad-dimensional torus,Bis the family of Borel
subsets ofXandμ=λis Lebesgue measure. For anyd×dmatrixA=(αjk)of
integers, letRA:X→Xbe the map defined byRAx=x′,where

ξ′j≡

∑d

k= 1

αjkξk mod 1(j= 1 ,...,d).

If detA=0thenRAis not measure-preserving, since the image ofRdunderAis con-
tained in a hyperplane ofRd. However, if detA=0thenRAis measure-preserving,
since each point ofXis the image underRAof|detA|distinct points ofX,anda
small regionBofXis the image underRAof|detA|disjoint regions, each with vol-
ume|detA|−^1 times that ofB. (This argument is certainly valid ifAis a diagonal
matrix, and the general case may be reduced to this by Proposition III.41.) ThusRAis
anendomorphismof the torusRd/Zdif and only ifAis nonsingular, and anautomor-
phismif and only if detA=±1.

Proposition 20The endomorphism RA:X →X of the d-dimensional torus X=
Rd/Zdis ergodic if and only if no eigenvalue of the nonsingular matrix A is a root of
unity.

Proof For anyn∈Zdwe have

e(n·RAx)=e(n·Ax)=e(Dn·x),

whereD=Atis the transpose ofA.
Suppose first thatA, and hence alsoD, has an eigenvalueωwhich is a root of unity:
ωp=1 for some positive integerp.Then(Dp−I)z=0 for some nonzero vectorz.
Moreover, sinceDis a matrix of integers, we may assume thatz=m∈Zd.Wemay
further assume thatDim=Djmfor 0≤i<j<p, by choosingpto have its least
possible value. If we put

f(x)=e(m·x)+e(m·Ax)+···+e(m·Ap−^1 x),

thenf(RAx)=f(x),butfis not constant a.e. HenceRAis not ergodic, by Proposi-
tion 18.
Suppose next thatRAis not ergodic. Then, by Proposition 18 again, there exists a
functionf∈L(X,B,λ)such thatf(RAx)=f(x)a.e., butf(x)is not constant a.e.
If the Fourier series off(x)is
∑

n∈Zd

cne(n·x),

then the Fourier series off(RAx)is

∑

n∈Zd

cne(n·Ax)=

∑

n∈Zd

cne(Dn·x)=

∑

n∈Zd

cD− (^1) ne(n·x)