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).
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