1.1. Equidistribution in tori 11
Exercise 1.1.13. Let
P (n1,...,nm) :=
i1,...,im≥0:i1+···+im≤s
αi1,...,im n11
i
. . .
nmmi
be a polynomial map from
Zm
to
Td
of degree s, where αi1,...,im
Td
are
coefficients. Show that the following are equivalent:
(i) P is asymptotically equidistributed on
Zm.
(ii) P is totally asymptotically equidistributed on
Zm.
(iii) There does not exist a non-zero k
Zd
such that k · αi1,...,im = 0
for all (i1,...,im) = 0.
Exercise 1.1.14 (Equidistribution for abelian multidimensional polynomial
sequences). Let T be a torus, and let P be a polynomial map from
Zm
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
Zm,
and P is periodic with respect
to some finite index sublattice of
Zm
(or equivalently, that all non-constant
coefficients of P are rational).
We give just one application of this multidimensional theory, that gives a
hint as to why the theory of equidistribution of polynomials may be relevant:
Exercise 1.1.15. Let T be a torus, let P be a polynomial map from Z
to T , let ε 0, and let k 1. Show that there exists positive integers
a, r 1 such that P (a),P (a + r),...,P (a + (k 1)r) all lie within ε of each
other. (Hint: Consider the polynomial map from
Z2
to T
k
that maps (a, r)
to (P (a),...,P (a +(k 1)r)). One can also use the one-dimensional theory
by freezing a and only looking at the equidistribution in r.)
1.1.2. Single-scale equidistribution theory. We now turn from the as-
ymptotic equidistribution theory to the equidistribution theory at a single
scale N. Thus, instead of analysing the qualitative distribution of infinite
sequence x: N X, we consider instead the quantitative distribution of
a finite sequence x: [N] X, where N is a (large) natural number and
[N] := {1,...,N}. To make everything quantitative, we will replace the
notion of a continuous function by that of a Lipschitz function. Recall that
the (inhomogeneous) Lipschitz norm f
Lip
of a function f : X R on a
metric space X = (X, d) is defined by the formula
f
Lip
:= sup
x∈X
|f(x)| + sup
x,y∈X:x=y
|f(x) f(y)|
d(x, y)
.
We also define the homogeneous Lipschitz semi-norm
f
˙
Lip
:= sup
x,y∈X:x=y
|f(x) f(y)|
d(x, y)
.
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