1.1. Equidistribution in tori 17

that V ∩

Zd

is a full rank sublattice of V , and is thus generated by dim(V )

independent generators.)

Proposition 1.1.17 (Single-scale equidistribution theorem for abelian poly-

nomial sequences). Let P be a polynomial map from Z to

Td

of some degree

s ≥ 0, and let F :

R+

→

R+

be an increasing function. Then there exists

an integer 1 ≤ M ≤ OF,s,d(1) and 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(M/N

i).

In particular, on the interval [N], Psmth is Lipschitz

with homogeneous norm Os,d(M/N).

(ii) (Pequi is equidistributed) There exists a subtorus T of

Td

of com-

plexity at most M and some dimension d , such that Pequi takes

values in T and is totally 1/F (M)-equidistributed on [N] in this

torus (after identifying this torus with

Td

using an invertible lin-

ear transformation of complexity at most M).

(iii) (Prat is rational) The coeﬃcients αi,rat of Prat are such that qαi,rat

= 0 for some 1 ≤ q ≤ M and all 0 ≤ i ≤ s. In particular, qPrat = 0

and Prat is periodic with period q.

If, furthermore, F is of polynomial growth, and more precisely F (M) ≤

KM

A

for some A, K ≥ 1, then one can take M

A,s,d

KOA,s,d(1).

Example 1.1.18. Consider the linear flow P (n) := (

√

2n, (

1

2

+

1

N

)n) mod

Z2

in

T2

on [N]. This flow can be decomposed into a smooth flow Psmth(n) :=

(0,

1

N

n) mod

Z2

with a homogeneous Lipschitz norm of O(1/N), an equidis-

tributed flow Pequi(n) := (

√

2n, 0) mod Z2 which will be δ-equidistributed

on the subtorus

T1

× {0} for a reasonably small δ (in fact one can take δ as

small as N

−c

for some small absolute constant c 0), and a rational flow

Prat(n) := (0,

1

2

n) mod

Z2,

which is periodic with period 2. This example

illustrates how all three components of this decomposition arise naturally in

the single-scale case.

Remark 1.1.19. Comparing this result with the asymptotically equidis-

tributed analogue in Example 1.1.7, we notice several differences. Firstly,

we now have the smooth component Psmth, which did not previously make

an appearance (except implicitly, as the constant term in P ). Secondly,

the equidistribution of the component Pequi is not infinite, but is the next

best thing, namely it is given by an arbitrary function F of the quantity M,

which controls the other components of the decomposition.