1.1. Equidistribution in tori 11

Exercise 1.1.13. Let

P (n1,...,nm) :=

i1,...,im≥0:i1+···+im≤s

αi1,...,im n11

i

. . .

nmmi

be a polynomial map from

Zm

to

Td

of degree s, where αi1,...,im ∈

Td

are

coeﬃcients. Show that the following are equivalent:

(i) P is asymptotically equidistributed on

Zm.

(ii) P is totally asymptotically equidistributed on

Zm.

(iii) There does not exist a non-zero k ∈

Zd

such that k · αi1,...,im = 0

for all (i1,...,im) = 0.

Exercise 1.1.14 (Equidistribution for abelian multidimensional polynomial

sequences). Let T be a torus, and let P be a polynomial map from

Zm

to

T of some degree s ≥ 0. Show that there exists a decomposition P = P +

P , where P , P are polynomials of degree s, P is totally asymptotically

equidistributed in a subtorus T of T on

Zm,

and P is periodic with respect

to some finite index sublattice of

Zm

(or equivalently, that all non-constant

coeﬃcients of P are rational).

We give just one application of this multidimensional theory, that gives a

hint as to why the theory of equidistribution of polynomials may be relevant:

Exercise 1.1.15. Let T be a torus, let P be a polynomial map from Z

to T , let ε 0, and let k ≥ 1. Show that there exists positive integers

a, r ≥ 1 such that P (a),P (a + r),...,P (a + (k − 1)r) all lie within ε of each

other. (Hint: Consider the polynomial map from

Z2

to T

k

that maps (a, r)

to (P (a),...,P (a +(k − 1)r)). One can also use the one-dimensional theory

by freezing a and only looking at the equidistribution in r.)

1.1.2. Single-scale equidistribution theory. We now turn from the as-

ymptotic equidistribution theory to the equidistribution theory at a single

scale N. Thus, instead of analysing the qualitative distribution of infinite

sequence x: N → X, we consider instead the quantitative distribution of

a finite sequence x: [N] → X, where N is a (large) natural number and

[N] := {1,...,N}. To make everything quantitative, we will replace the

notion of a continuous function by that of a Lipschitz function. Recall that

the (inhomogeneous) Lipschitz norm f

Lip

of a function f : X → R on a

metric space X = (X, d) is defined by the formula

f

Lip

:= sup

x∈X

|f(x)| + sup

x,y∈X:x=y

|f(x) − f(y)|

d(x, y)

.

We also define the homogeneous Lipschitz semi-norm

f

˙

Lip

:= sup

x,y∈X:x=y

|f(x) − f(y)|

d(x, y)

.