Number Theory: An Introduction to Mathematics

1 Uniform Distribution 453

It follows that each point of the sequence(nx)lies on some hyperplanem·y =h,
whereh ∈ Z. Without loss of generality, supposeμ 1 =0. Then no point of the
d-dimensional unit cube which is sufficiently close to the point(| 2 μ 1 |−^1 , 0 ,..., 0 )
lies on such a hyperplane.
We now return to the one-dimensional case. Weyl used Theorem 2 to prove, not
only Proposition 3, but also a deeper result concerning the uniform distribution of
the sequence(f(n)),wherefis a polynomial of any positive degree. We will derive
Weyl’s result by a more general argument due to van der Corput (1931), based on the
following inequality:

Lemma 4Ifζ 1 ,...,ζNare arbitrary complex numbers then, for any positive integer
M≤N,

M^2

∣

∣

∣

∣

∑N

n= 1

ζn

∣

∣

∣

∣

2

≤M(M+N− 1 )

∑N

n= 1

|ζn|^2 + 2 (M+N− 1 )

M∑− 1

m= 1

(M−m)

∣

∣

∣

∣

N∑−m

n= 1

ζnζn+m

∣

∣

∣

∣.

Proof Putζn=0ifn≤0orn>N. Then it is easily verified that

M

∑N

n= 1

ζn=

M+∑N− 1

h= 1

(M∑− 1

k= 0

ζh−k

)

.

Applying Schwarz’s inequality (Chapter I,§4), we get

M^2

∣

∣

∣

∣

∑N

n= 1

ζn

∣

∣

∣

∣

2

≤(M+N− 1 )

M+∑N− 1

h= 1

∣

∣

∣

∣

M∑− 1

k= 0

ζh−k

∣

∣

∣

∣

2

=(M+N− 1 )

M+∑N− 1

h= 1

M∑− 1

j,k= 0

ζh−kζh−j.

On the right side any term|ζn|^2 occurs exactlyMtimes, namely forh−k=h−j=n.
Atermζnζn+morζnζn+m,wherem>0, occurs only ifm<Mand then it occurs
exactlyM−mtimes. Thus the right side is equal to

M(M+N− 1 )

∑N

n= 1

|ζn|^2 +(M+N− 1 )

M∑− 1

m= 1

(M−m)

N∑−m

n= 1

(ζnζn+m+ζnζn+m).

The lemma follows.

Corollary 5If(ξn)is a real sequence such that, for each positive integer m,

N−^1

∑N

n= 1

e(ξn+m−ξn)→ 0 as N→∞,

then

N−^1

∑N

n= 1

e(ξn)→ 0 as N→∞.