1.1. Equidistribution in tori 25

P = Psmth + Pequi,Prat, then Pi and Pi differ by expressions taking values

in T ).

Exercise 1.1.41 (Recurrence). Let P be a polynomial map from ∗Z to

∗Td

of some standard degree s, and let N be an unbounded natural number.

Show that for every standard ε 0 and every n0 ∈ N, we have

|{n ∈ [N] : P (n) − P (n0) ≤ ε}| N

and more generally

|{r ∈ [−N, N] : P (n0 + jr) − P (n0) ≤ ε for j = 0, 1,...,k − 1}| N

for any standard k.

As before, there are also multidimensional analogues of this theory. We

shall just state the main results without proof:

Definition 1.1.25 (Multidimensional equidistribution). Let X be a stan-

dard compact metric space, let N be an unbounded limit natural number,

let m ≥ 1 be standard, and let x:

[N]m

→ ∗X be a limit function. We

say that x is equidistributed with respect to a (standard) Borel probability

measure μ on X if one has

stEn∈[N]m 1B(n/N)f(x(n)) = mes(Ω)

X

f dμ

for every standard box B ⊂ [0,

1]m

and for all standard continuous functions

f ∈ C(X).

We say that x is totally equidistributed relative to μ if the sequence

n → x(qn + r) is equidistributed on

[N/q]d

for every standard q 0 and

r ∈

Zm

(extending x arbitrarily outside [N] if necessary).

Remark 1.1.26. One can replace the indicators 1B by many other classes,

such as indicators of standard convex sets, or standard open sets whose

boundary has measure zero, or continuous or Lipschitz functions.

Theorem 1.1.27 (Multidimensional ultralimit equidistribution theorem for

abelian polynomial sequences). Let m, d, s ≥ 0 be standard integers, and let

P be a polynomial map from

∗Zm

to

∗Td

of degree s. Let N be an unbounded

natural number. Then there exists a decomposition

P = Psmth + Pequi + Prat

into polynomials of degree s, where

(i) (Psmth is smooth) The

ith

coeﬃcient αi,smth of Psmth has size

O(N

−|i|)

for every multi-index i = (i1,...,im). In particular,

on the interval [N], Psmth is Lipschitz with homogeneous norm

O(1/N).