488 XI Uniform Distribution and Ergodic Theory
i.e.z 1 =zn+ 1 =z 2 n+ 1 =···=zpn+ 1 .Sincez∈X, there is a positive integermsuch
that d(τmx,z)< 2 −pn−^1 ,i.e.xm+i=zifor 1≤i≤pn+1. It follows that
xm+ 1 =xm+n+ 1 =···=xm+pn+ 1.Thus for every positive integerpthere is a setSj(p)which contains an arithmetic
progression of lengthp. Since there are onlyrpossible values forj(p), one of the sets
Sjmust contain arithmetic progressions of arbitrary finite length.
A far-reaching generalization of van der Waerden’s theorem has been given by
Hales and Jewett (1963). LetA={a 1 ,...,aq}be a finite set and letAnbe the set
of alln-tuples with elements fromA.AsetW={w^1 ,...,wq}⊆Anofqn-tuples
wk=(wk 1 ,...,wkn)is said to be acombinatorial lineif there exists a partition
{ 1 ,...,n}=I∪J, I∩J=∅,such that
wik=ak(k= 1 ,...,q) fori∈I; w^1 j=···=wqj forj∈J.The Hales–Jewett theorem says that, for any positive integerr, there exists a posi-
tive integerN=N(q,r)such that, ifANis partitioned intorclasses, then at least one
of these classes contains a combinatorial line.
If one takesA={ 0 , 1 ,...,q− 1 }and interpretsAnas the set of expansions to base
qof all non-negative integers less thanqn, then a combinatorial line is an arithmetic
progression. On the other hand, if one takesA=Fqto be a finite field withqelements
and interpretsAnas then-dimensional vector spaceFnq, then a combinatorial line is
an affine line. The interesting feature of theHales–Jewett theorem is that it is purely
combinatorial and does not involve any notion of addition.
6 FurtherRemarks
Uniform distribution and discrepancy are thoroughly discussed in Kuipers and Nieder-
reiter [30]. For later results, see Drmota and Tichy [13]. Since these two books have
extensive bibliographies, we will be sparing with references. However, it would be re-
miss not to recommend the great paper of Weyl [52], which remains as fresh as when
it was written.
Lemma 0 is often attributed to Polya (1920), but it was already proved by Buchanan
and Hildebrandt [9].
Fej ́er’s proof that continuous periodic functions can be uniformly approximated by
trigonometric polynomials is given in Dym and McKean [15]. The theorem also fol-
lows directly from the the theorem of Weierstrass (1885) on the uniform approximation
of continuous functions by ordinary polynomials. A remarkable generalization of both
results was given by Stone (1937); see Stone [49]. The ‘Stone–Weierstrass theorem’ is
also proved in Rudin [44], for example.
Chen [11] gives a quantitative version of Kronecker’s theorem of a different type
from Proposition 3′.