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:
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