Number Theory: An Introduction to Mathematics

468 XI Uniform Distribution and Ergodic Theory

and hence

τ(∑x)− 1

k= 0

f ̄M(Tkx)≤

τ(∑x)− 1

k= 0

f ̃(Tkx)+τ(x)ε.

To estimate the sum

∑L− 1

k= 0 f ̄M(T

kx)for anyL>N, we partition it into blocks of

the form

τ(∑y)− 1

k= 0

f ̄M(Tky)

and a remainder block. More precisely, define inductively

n 0 (x)= 0 , nk(x)=nk− 1 (x)+τ(Tnk−^1 x)(k= 1 , 2 ,...)

and definehbynh(x)<L≤nh+ 1 (x).Then

n (^1) ∑(x)− 1
k= 0
f ̄M(Tkx)≤
n (^1) ∑(x)− 1
k= 0
f ̃(Tkx)+τ(x)ε,
n (^2) ∑(x)− 1
k=n 1 (x)
f ̄M(Tkx)≤
n (^2) ∑(x)− 1
k=n 1 (x)
f ̃(Tkx)+τ(Tn^1 x)ε,

···

nh∑(x)− 1

k=nh− 1 (x)

f ̄M(Tkx)≤

nh∑(x)− 1

k=nh− 1 (x)

f ̃(Tkx)+τ(Tnh−^1 x)ε.

Sincenh(x)<L, we obtain by addition

nh∑(x)− 1

k= 0

f ̄M(Tkx)≤

nh∑(x)− 1

k= 0

f ̃(Tkx)+Lε.

On the other hand, sinceL≤nh+ 1 (x)≤nh(x)+N,wehave

L∑− 1

k=nh(x)

f ̄M(Tkx)≤NM.

Sincef ̃≥0, it follows that

L∑− 1

k= 0

f ̄M(Tkx)≤

L∑− 1

k= 0

f ̃(Tkx)+Lε+NM.

Dividing byLand integrating overX, we obtain
∫

X

f ̄Mdμ≤

∫

X

f ̃dμ+ε+NM/L,