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?
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