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

+

.