490 XI Uniform Distribution and Ergodic Theory
Applications of ergodic theory to classical mechanics are discussed in the books of
Arnold and Avez [1] and Katok and Hasselblatt [25]. For connections between ergodic
theory and the ‘3x+1 problem’, see Lagarias [31].
Ergodic theory has been used to generalize considerably some of the results on lat-
tices in Chapter VIII. Alatticein a locally compact groupGis a discrete subgroupΓ
such that theG-invariant measure of the quotient spaceG/Γis finite. (In Chapter VIII,
G=RnandΓ =Zn.) Zimmer [54] gives a good introduction to the results which
have been obtained in this area.
An attractive account of the work of Furstenberg and his collaborators is given in
Furstenberg [17]. See also Grahamet al.[20] and the book of Petersen cited above.
The discovery of van der Waerden’s theorem is described in van der Waerden [50]. For
a recent direct proof, see Mills [34].
The direct proofs reduce the theorem to an equivalent finite form:for any positive
integer p, there exists a positive integer N such that, whenever the set{ 1 , 2 ,...,N}
is partitioned into two subsets, at least one subset contains an arithmetic progression
of length p. The original proofs provided an upper bound for the least possible value
N(p)ofN, but it was unreasonably large. Some progress towards obtaining reasonable
upper bounds has recently been made by Shelah [47] and Gowers [19].
The Hales–Jewett theorem is proved, and then extensively generalized, in
Bergelson and Leibman [5]. Furstenberg and Katznelson [18] prove a density ver-
sion of the Hales–Jewett theorem, analogous to Szemeredi’s density version of van der
Waerden’s theorem.
7 SelectedReferences
[1] V.I. Arnold and A. Avez,Ergodic problems of classical mechanics, Benjamin, New York,
1968.
[2] K.I. Babenko, On a problem of Gauss,Soviet Math. Dokl. 19 (1978), 136–140.
[3] J. Beck, Probabilisticdiophantine approximation, I. Kronecker sequences,Ann. of Math.
140 (1994), 451–502.
[4] J. Beck and W.W.L. Chen,Irregularities of distribution, Cambridge University Press, 1987.
[5] V. Bergelson and A. Leibman, Set polynomials and polynomial extension of the
Hales–Jewett theorem,Ann. of Math. 150 (1999), 33–75.
[6] P. Billingsley,Probabilityand measure, 3rd ed., Wiley, New York, 1995.
[7] P. Billingsley,Ergodic theory and information, reprinted, Krieger, Huntington, N.Y., 1978.
[8] G. Brown and W. Moran, Schmidt’s conjecture on normality for commuting matrices,
Invent. Math. 111 (1993), 449–463.
[9] H.E. Buchanan and H.T. Hildebrandt, Note on the convergence of a sequence of functions
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(2000), 691–705.
[12] I.P. Cornfeld, S.V. Fomin and Ya. G. Sinai,Ergodic theory, Springer-Verlag, New York,
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[13] M. Drmota and R.F. Tichy,Sequences, discrepancies and applications, Lecture Notes in
Mathematics 1651 , Springer, Berlin, 1997.