Number Theory: An Introduction to Mathematics

484 XI Uniform Distribution and Ergodic Theory

points. Poincar ́e’s proof was inevitably incomplete, since at the time measure theory
did not exist. However, Carath ́eodory (1919) showed that his argument could be made
rigorous with the aid of Lebesgue measure:

Proposition 25Let T:X→X be a measure-preserving transformation of the prob-
ability space ( X,B,μ). Then almost all points of any B∈Breturn to B infinitely
often, i.e. for each x∈B, apart from a set ofμ-measure zero, there exists an increasing
sequence (nk) of positive integers such that Tnkx∈B(k= 1 , 2 ,...).
Furthermore, ifμ(B)> 0 ,thenμ(B∩T−nB)> 0 for infinitely many n≥ 1.

Proof For anyN≥0, putBN=

⋃

n≥NT

−nB.Then

A:=∩

N≥ 0

BN

is the set of all pointsx∈Xsuch thatTnx∈Bfor infinitely many positive integers
n.SinceBN+ 1 =T−^1 BN,wehaveμ(BN+ 1 )=μ(BN)and henceμ(BN)=μ(B 0 )
for allN≥1. SinceBN+ 1 ⊆BN, it follows that

μ(A)= lim

N→∞

μ(BN)=μ(B 0 ).

SinceA⊆B 0 , this implies

μ(B 0 \A)=μ(B 0 )−μ(A)= 0

and hence, sinceB⊆B 0 ,μ(B\A)=0.
This proves the first statement of the proposition. Ifμ(B∩T−nB)=0forall
n≥m,thenμ(B∩BN)=0forallN≥mand hence

μ(B∩A)= lim

N→∞

μ(B∩BN)= 0.

Consequently

μ(B)=μ(B\A)+μ(B∩A)= 0 ,

which proves the second statement of the proposition.

Furstenberg (1977) extended Proposition 25 in the following way:
Let T be a measure-preserving transformation of the probability space(X,B,μ).
If B∈Bwithμ(B)> 0 and if p≥2,thenμ(B∩T−nB∩···∩T−(p−^1 )nB)> 0
for some n≥1.
His proof of this theorem made heavy use of ergodic theory and, in particular,
of a new structure theory for measure-preserving transformations. From his theorem
he was able to deduce quite easily a result for which Szemeredi (1975) had given a
complicated combinatorial proof:
Let S be a subset of the setNof positive integers which has positive upper density;
i.e., for someα∈( 0 , 1 ),there exist arbitrarily long intervals I⊆Ncontaining at least
α|I|elements of S. Then S contains arithmetic progressions of arbitrary finite length.