Number Theory: An Introduction to Mathematics

452 XI Uniform Distribution and Ergodic Theory

These results can be immediately extended to higher dimensions. A sequence
(xn)of vectors inRdis said to beuniformly distributed mod1 if, for all vectors
a=(α 1 ,...,αd)andb=(β 1 ,...,βd)with 0≤αk<βk≤ 1 (k= 1 ,...,d),

φa,b(N)/N→

∏d

k= 1

(βk−αk) asN→∞,

wherexn=(ξn(^1 ),...,ξn(d))andφa,b(N)is the number of positive integersn≤N
such thatαk ≤{ξn(k)} <βk for everyk ∈{ 1 ,...,d}.LetIdbe the set of all
x =(ξ(^1 ),...,ξ(d))such that 0≤ξ(k)≤ 1 (k = 1 ,...,d)and, for an arbitrary
vectorx=(ξ(^1 ),...,ξ(d)), put

{x}=({ξ(^1 )},...,{ξ(d)}).

Then Theorems 1 and 2 have the following generalizations:

Theorem 1′ A sequence(xn)of vectors inRdis uniformly distributed mod 1 if and
only if, for every function f:Id→Cwhich is Riemann integrable,

N−^1

∑N

n= 1

f({xn})→

∫

I

···

∫

I

f(t 1 ,...,td)dt 1 ···dtd as N→∞.

Theorem 2′ A sequence(xn)of vectors inRdis uniformly distributed mod 1 if and
only if, for every nonzero vector m=(μ 1 ,...,μd)∈Zd,

N−^1

∑N

n= 1

e(m·xn)→ 0 as N→∞,

where m·xn=μ 1 ξn(^1 )+···+μdξn(d).

Proposition 3 can also be generalized in the following way:

Proposition 3′If x=(ξ(^1 ),...,ξ(d))is any vector inRdsuch that 1 ,ξ(^1 ),...,ξ(d)
are linearly independent over the fieldQof rational numbers, then the sequence(nx)
is uniformly distributed mod 1.

In particular, the sequence({nx}) = ({nξ(^1 )},...,{nξ(d)})is dense in the
d-dimensional unit cube if 1,ξ(^1 ),...,ξ(d)are linearly independent over the fieldQ
of rational numbers. This much weaker assertion had already been proved before Weyl
by Kronecker (1884).
It is easily seen that the linear independence of 1,ξ(^1 ),...,ξ(d)over the fieldQof
rational numbers is also necessary for the sequence({nx})to be dense in the
d-dimensional unit cube and,a fortiori, for the sequence(nx)to be uniformly dis-
tributed mod 1. For if 1,ξ(^1 ),...,ξ(d)are linearly dependent overQthere exists a
nonzero vectorm=(μ 1 ,...,μd)∈Zdsuch that

m·x=μ 1 ξ(^1 )+···+μdξ(d)∈Z.