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.
Previous Page Next Page