1.1. Equidistribution in tori 13
Note that the above exercise does not specify the exact relationship
between δ and N when one is given an asymptotically equidistributed se-
quence x(1),x(2),... ; this relationship is the additional piece of information
provided by single-scale equidistribution that is not present in asymptotic
equidistribution.
It turns out that much of the asymptotic equidistribution theory has
a counterpart for single-scale equidistribution. We begin with the Weyl
criterion.
Proposition 1.1.13 (Single-scale Weyl equidistribution criterion). Let x1,
x2,...,xN be a sequence in
Td,
and let 0 δ 1.
(i) If x1,...,xN is δ-equidistributed, and k
Zd\{0}
has magnitude
|k|
δ−c,
then one has
|En∈[N]e(k · xn)|
d
δc
if c 0 is a small enough absolute constant.
(ii) Conversely, if x1,...,xN is not δ-equidistributed, then there exists
k
Zd\{0}
with magnitude |k|
d
δ−Cd
, such that
|En∈[N]e(k · xn)|
d
δCd
for some Cd depending on d.
Proof. The first claim is immediate as the function x e(k · x) has mean
zero and Lipschitz constant Od(|k|), so we turn to the second claim. By
hypothesis, (1.6) fails for some Lipschitz f. We may subtract off the mean
and assume that
Td
f = 0; we can then normalise the Lipschitz norm to be
one; thus we now have
|En∈[N]f(xn)| δ.
We introduce a summation parameter R N, and consider the Fej´ er partial
Fourier series
FRf(x) :=
k∈Zd
mR(k)
ˆ(k)e(k
f · x)
where
ˆ(k)
f are the Fourier coefficients
ˆ(k)
f :=
Td
f(x)e(−k · x) dx
and mR is the Fourier multiplier
mR(k1,...,kd) :=
d
j=1
1
|kj|
R
+
.
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