12 1. Higher order Fourier analysis

Definition 1.1.11. Let X = (X, d) be a compact metric space, let δ 0,

let μ be a probability measure on X. A finite sequence x: [N] → X is said

to be δ-equidistributed relative to μ if one has

(1.6) |En∈[N]f(x(n)) −

X

f dμ| ≤ δ f

Lip

for all Lipschitz functions f : X → R.

We say that the sequence x1,...,xN ∈ X is totally δ-equidistributed

relative to μ if one has

|En∈P f(x(n)) −

X

f dμ| ≤ δ f

Lip

for all Lipschitz functions f : X → R and all arithmetic progressions P in

[N] of length at least δN.

In this section, we will only apply this concept to the torus

Td

with

the Haar measure μ and the metric inherited from the Euclidean metric.

However, in subsequent sections we will also consider equidistribution in

other spaces, most notably on nilmanifolds.

Exercise 1.1.16. Let x(1),x(2),x(3),... be a sequence in a metric space

X = (X, d), and let μ be a probability measure on X. Show that the

sequence x(1),x(2),... is asymptotically equidistributed relative to μ if

and only if, for every δ 0, x(1),...,x(N) is δ-equidistributed relative

to μ whenever N is suﬃciently large depending on δ, or equivalently if

x(1),...,x(N) is δ(N)-equidistributed relative to μ for all N 0, where

δ(N) → 0 as N → ∞. (Hint: You will need the Arzel´ a-Ascoli theorem.)

Similarly, show that x(1),x(2),... is totally asymptotically equidis-

tributed relative to μ if and only if, for every δ 0, x(1),...,x(N) is totally

δ-equidistributed relative to μ whenever N is suﬃciently large depending on

δ, or equivalently if x(1),...,x(N) is totally δ(N)-equidistributed relative

to μ for all N 0, where δ(N) → 0 as N → ∞.

Remark 1.1.12. More succinctly, (total) asymptotic equidistribution of

x(1),x(2),... is equivalent to (total) oN→∞(1)-equidistribution of x(1),...,

x(N) as N → ∞, where on→∞(1) denotes a quantity that goes to zero

as N → ∞. Thus we see that asymptotic notation such as on→∞(1) can

eﬃciently conceal a surprisingly large number of quantifiers.

Exercise 1.1.17. Let N0 be a large integer, and let x(n) := n/N0 mod 1 be

a sequence in the standard torus T = R/Z with Haar measure. Show that

whenever N is a positive multiple of N0, then the sequence x(1),...,x(N)

is O(1/N0)-equidistributed. What happens if N is not a multiple of N0?

If, furthermore, N ≥ N0

2,

show that x(1),...,x(N) is O(1/

√

N0)-equi-

distributed. Why is a condition such as N ≥ N0

2

necessary?