448 XI Uniform Distribution and Ergodic Theory
For any real numberξ,let ξ denote again the greatest integer≤ξand let{ξ}=ξ− ξ denote thefractional partofξ. We are going to prove that, ifξis irrational, then the
sequence({nξ})of the fractional parts of the multiples ofξisdensein the unit interval
I=[0,1], i.e. every point ofIis a limit point of the sequence.
It is sufficient to show that the pointszn=e^2 πinξ(n= 1 , 2 ,...)are dense on the
unit circle. Sinceξis irrational, the pointsznare all distinct andzn=±1. Conse-
quently they have a limit point on the unit circle. Thus, for any givenε>0, there exist
positive integersm,rsuch that
|zm+r−zm|<ε.But
|zm+r−zm|=|zr− 1 |=|zn+r−zn| for everyn∈N.If we writezr=e^2 πiθ,where0<θ<1, thenzkr=e^2 πikθ(k= 1 , 2 ,...).Definethe
positive integerNby 1/(N+ 1 )<θ< 1 /N. Then the pointszr,z 2 r,...,zNrfollow
one another in order on the unit circle and every point of the unit circle is distant less
thanεfrom one of these points.
It may be asked if the sequence({nξ})is not only dense inI, but also spends
‘the right amount of time’ in each subinterval ofI. To make the question precise we
introduce the following definition:
A sequence(ξn)of real numbers is said to beuniformly distributed mod1 if, for
allα,βwith 0≤α<β≤1,
φα,β(N)/N→β−α asN→∞,whereφα,β(N)is the number of positive integersn≤Nsuch thatα≤{ξn}<β.
In this definition we need only require thatφ 0 ,α(N)/N→αfor everyα∈( 0 , 1 ),
since
φα,β(N)=φ 0 ,β(N)−φ 0 ,α(N)and hence
|φα,β(N)/N−(β−α)|≤|φ 0 ,β(N)/N−β|+|φ 0 ,α(N)/N−α|.It follows from Lemma 0, withfn(t)=φ 0 ,t(n)/nandf(t)=t, that the sequence(ξn)
is uniformly distributed mod 1 if and only if
φα,β(N)/N→β−α asN→∞uniformlyfor allα,βwith 0≤α<β≤1.
It was first shown by Bohl (1909) that, ifξis irrational, the sequence(nξ)is uni-
formly distributed mod 1 in the sense of our definition. Later Weyl (1914,1916) estab-
lished this result by a less elementary, but much more general argument, which was
equally applicable to multi-dimensional problems. The following two theorems, due to
Weyl, replace the problem of showing that a sequence is uniformly distributed mod 1
by a more tractable analytic problem.