1.1. Equidistribution in tori 9

Proof. We induct on s. For s = 1 this follows from Exercise 1.1.5. Now

suppose that s 1, and that the claim has already been proven for smaller

values of s. For any positive integer h, we observe that P (n + h) − P (n)

is a polynomial of degree s − 1 in n with leading coeﬃcient

shαsns−1.

As

αs is irrational, shαs is irrational also, and so by the induction hypothesis,

P (n + h) − P (n) is asymptotically equidistributed. The claim now follows

from Corollary 1.1.7.

Exercise 1.1.6. Let P (n) =

αsns

+ · · · + α0 be a polynomial of degree s in

Td.

Show that the following are equivalent:

(i) P is asymptotically equidistributed on N.

(ii) P is totally asymptotically equidistributed on N.

(iii) P is totally asymptotically equidistributed on Z.

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

Zd

such that k · α1 = · · · =

k · αs = 0.

(Hint: It is convenient to first use Corollary 1.1.3 to reduce to the one-

dimensional case.)

This gives a polynomial variant of the equidistribution theorem:

Exercise 1.1.7 (Equidistribution theorem for abelian polynomial sequences).

Let T be a torus, and let P be a polynomial map from Z 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 Z, and P is periodic (or equivalently, that all non-

constant coeﬃcients of P are rational).

In particular, we see that polynomial sequences in a torus are equidis-

tributed with respect to a finite combination of Haar measures of cosets of

a subtorus. Note that this finite combination can have multiplicity; for in-

stance, when considering the polynomial map n → (

√

2n,

1

3

n2)

mod

Z2,

it

is not hard to see that this map is equidistributed with respect to 1/3 times

the Haar probability measure on (T) × {0 mod Z}, plus 2/3 times the Haar

probability measure on (T) × {

1

3

mod Z}.

Exercise 1.1.7 gives a satisfactory description of the asymptotic equidis-

tribution of arbitrary polynomial sequences in tori. We give just one example

of how such a description can be useful:

Exercise 1.1.8 (Recurrence). Let T be a torus, let P be a polynomial map

from Z to T , and let n0 be an integer. Show that there exists a sequence nj

of positive integers going to infinity such that P (nj) → P (n0).