Number Theory: An Introduction to Mathematics

1 Uniform Distribution 457

It may be noted that the matrixAin Proposition 9 is necessarily non-singular.
For if detA =0, there exists a nonzero vectorz∈Zdsuch thatAtz =0. Then,
for anyx ∈Rdand any positive integern,e(z·Anx)=e((At)nz·x)=1. Thus
N−^1

∑N

n= 1 e(z·A

nx)=1 and therefore, by Theorem 2′, the sequenceAnxis not

uniformly distributed mod 1.
Further examples of uniformly distributed sequences are provided by the following
result, which is due to Fej ́er (c. 1924):

Proposition 10Let(ξn)be a sequence of real numbers such thatηn :=ξn+ 1 −ξn
tends to zero monotonically as n→∞.Then(ξn)is uniformly distributed mod 1 if
n|ηn|→∞as n→∞.

Proof By changing the signs of allξnwe may restrict attention to the case where the
sequence(ηn)is strictly decreasing. For any real numbersα,βwe have

|e(α)−e(β)− 2 πi(α−β)e(β)|=|e(α−β)− 1 − 2 πi(α−β)|

= 4 π^2

∣

∣

∣

∣

∫α−β

0

(α−β−t)e(t)dt

∣

∣

∣

∣

≤ 4 π^2

∣

∣

∣

∣

∫α−β

0

(α−β−t)dt

∣

∣

∣

∣

= 2 π^2 (α−β)^2.

If we takeα=hξn+ 1 andβ=hξn,wherehis any nonzero integer, this yields

|e(hξn+ 1 )/ηn−e(hξn)/ηn− 2 πihe(hξn)|≤ 2 π^2 h^2 ηn

and hence

|e(hξn+ 1 )/ηn+ 1 −e(hξn)/ηn− 2 πihe(hξn)|≤ 1 /ηn+ 1 − 1 /ηn+ 2 π^2 h^2 ηn.

Takingn= 1 ,...,Nand adding, we obtain

∣

∣

∣

∣^2 πh

∑N

n= 1

e(hξn)

∣

∣

∣

∣≤^1 /ηN+^1 +^1 /η^1 +

∑N

n= 1

( 1 /ηn+ 1 − 1 /ηn)+ 2 π^2 h^2

∑N

n= 1

ηn

= 2 /ηN+ 1 + 2 π^2 h^2

∑N

n= 1

ηn.

Thus

N−^1

∣

∣

∣

∣

∑N

n= 1

e(hξn)

∣

∣

∣

∣≤(π|h|NηN+^1 )

− (^1) +π|h|N− 1

∑N

n= 1

ηn.

But the right side of this inequality tends to zero asN→∞,sinceNηN →∞and
ηN→0.