14 1. Higher order Fourier analysis

Standard Fourier analysis shows that we have the convolution representation

FRf(x) =

Td

f(y)KR(x − y)

where KR is the Fej´ er kernel

KR(x1,...,xd) :=

d

j=1

1

R

sin(πRxj)

sin(πxj)

2

.

Using the kernel bounds

Td

KR = 1

and

|KR(x)|

d

d

j=1

R(1 + R xj

T)−2,

where x

T

is the distance from x to the nearest integer, and the Lipschitz

nature of f, we see that

FRf(x) = f(x) + Od(1/R).

Thus, if we choose R to be a suﬃciently small multiple of 1/δ (depending

on d), one has

|En∈[N]FRf(xn)| δ

and thus by the pigeonhole principle (and the trivial bound

ˆ(k)

f = O(1)

and

ˆ(0)

f = 0) we have

|En∈[N]e(k · xn)|

d

δOd(1)

for some non-zero k of magnitude |k|

d

δ−Od(1),

and the claim follows.

There is an analogue for total equidistribution:

Exercise 1.1.18. Let x1,x2,...,xN be a sequence in

Td,

and let 0 δ 1.

(i) If x1,...,xN is totally δ-equidistributed, k ∈

Zd\{0}

has magnitude

|k| ≤

δ−cd

, and a is a rational of height at most

δ−cd

, then one has

|En∈[N]e(k · xn)e(an)|

d

δcd

if cd 0 is a small enough constant depending only on d.

(ii) Conversely, if x1,...,xN is not totally δ-equidistributed, then there

exists k ∈

Zd\{0}

with magnitude |k|

d

δ−Cd

, and a rational a of

height

Od(δ−Cd

), such that

|En∈[N]e(k · xn)e(an)|

d

δCd

for some Cd depending on d.

This gives a version of Exercise 1.1.5: