Number Theory: An Introduction to Mathematics

6 Further Remarks 489

The converse of Proposition 10 is proved by Kemperman [27]. For the history of
the problem of mean motion, and generalizations to almost periodic functions, see
Jessen and Tornehave [24]. Methods for estimating exponential sums were developed
in connection with the theory of uniform distribution, but then found other applica-
tions. See Chandrasekharan [10] and Graham and Kolesnik [21].
For applications of discrepancy to numerical integration, see Niederreiter [36, 37].
For the basic properties of functions ofbounded variation and the definition of total
variation see, for example, Riesz and Sz.-Nagy [42].
Sharper versions of the original Erd ̋os–Turan inequality are proved by Niederreiter
and Philipp [38] and in Montgomery [35]. The discrepancy of the sequence({nα}),
whereαis an irrational number whose continued fraction expansion has bounded
partial quotients (i.e., isbadly approximable), is discussed by Dupain and S ́os [14].
The discrepancy of the sequence ({nα}), whereα∈Rd, has been deeply studied by
Beck [3]. The work of Roth, Schmidt and others is treated in Beck and Chen [4].
For accounts of measure theory, see Billingsley [6], Halmos [22], Lo`eve [32]
and Saks [46]. More detailed treatments of ergodic theory are given in the books of
Petersen [39], Walters [51] and Cornfeldet al.[12]. The prehistory of ergodic theory
is described by the Ehrenfests [16]. However, they do not refer to the paper of Poincar ́e
(1894), which is reproduced in [41].
The proof of Birkhoff’s ergodic theorem given here follows Katznelson and
Weiss [26]. A different proof is given in the book of Walters.
Many other ergodic theorems besides Birkhoff’s are discussed in Krengel [29]. We
mention only thesubadditive ergodic theoremof Kingman (1968): ifTis a measure-
preserving transformation of the probability space (X,B,μ)andif(gn) is a sequence
of functions inL(X,B,μ)such that infnn−^1

∫

Xgndμ>−∞and, for allm,n≥1,

gn+m(x)≤gn(x)+gm(Tnx)a.e.,

thenn−^1 gn(x)→g∗(x)a.e., whereg∗(Tx)=g∗(x)a.e.,g∗∈L(X,B,μ)and
∫

X

g∗dμ= lim

n→∞

n−^1

∫

X

gndμ=inf

n

n−^1

∫

X

gndμ.

Birkhoff’s ergodic theorem may be regarded as a special case by taking∑ gn(x)=
n− 1
k= 0 f(T

kx). A simple proof of Kingman’s theorem is given by Steele [48]. For

applications of Kingman’s theorem to percolation processes and products of random
matrices, see Kingman [28]. The multiplicative ergodic theorem of Oseledets is de-
rived from Kingman’s theorem by Ruelle [45].
The book of Kuipers and Niederreiter cited above has an extensive discussion of
normal numbers. For normality with respect to a matrix, see also Brown and Moran [8].
Proofs of Gauss’s statement on the continued fraction map are contained in the
books by Billingsley [7] and Rockett and Szusz [43]. For more recent work, see
Wirsing [53], Babenko [2] and Mayer [33]. For the deviation of( 1 /n)logqn(ξ)
from its (a.e.) limiting valueπ^2 /(12 log 2)there are analogues of the central limit
theorem and the law of the iterated logarithm; see Philipp and Stackelberg [40]. For
higher-dimensional generalizations of Gauss’s invariant measure, see Hardcastle and
Khanin [23].