Number Theory: An Introduction to Mathematics

462 XI Uniform Distribution and Ergodic Theory

As an application of Proposition 15 we prove

Proposition 16Ifξ 1 ,...,ξNare points of the unit interval I with discrepancy D∗N
then, for any integer h= 0 ,

∣

∣

∣

∣N

− 1

∑N

n= 1

e(hξn)

∣

∣

∣

∣≤^4 |h|D

∗

N.

Proof We can write

N−^1

∑N

n= 1

e(hξn)=ρe(α),

whereρ≥0andα∈I. Thus

ρ=N−^1

∑N

n= 1

e(hξn−α).

Adding this relation to its complex conjugate, we obtain

ρ=N−^1

∑N

n= 1

cos 2π(hξn−α).

The result follows by applying Proposition 15 to the functionf(t)=cos 2π(ht−α),
which has bounded variation onIwith total variation

∫

I|f

′(t)|dt= 4 |h|. 

An inequality in the opposite direction to Proposition 16 was obtained by Erd ̋os
and Turan (1948) who showed that, for any positive integerm,

D∗N≤C

(

m−^1 +

∑m

h= 1

h−^1

∣∣

∣

∣N

− 1

∑N

n= 1

e(hξn)

∣∣

∣

∣

)

,

where the positive constantCis independent ofm,Nand theξ’s. Niederreiter and
Philipp (1973) showed that one can takeC=4. Furthermore they generalized the
result and simplified the proof.
The connection between these results and the theory of uniform distribution is
close at hand. Let(ξn)be an arbitrary sequence of real numbers and letδNdenote the
discrepancy of the fractional parts{ξ 1 },...,{ξN}. By the remark after the definition of
uniform distribution in§1,the sequence(ξn)is uniformly distributed mod 1 if and only
ifδN→0asN→∞. It follows from Proposition 16 and the inequality of Erd ̋os and
Turan (in whichmmay be arbitrarily large) thatδN→0asN→∞if and only if,
for every integerh=0,

N−^1

∑N

n= 1

e(hξn)→0asN→∞.

This provides a new proof of Theorem 2. Furthermore, from bounds for the exponential
sums we can obtain estimates for the rapidity with whichδNtends to zero.