# Equidistribution of Polynomial maps on locally compact groups

Tom Meyerovitch  Ben Gurion University of the Negev. Departement of
Mathematics. Be’er Sheva, 8410501, Israel

###### Abstract.

We formulate and prove two generalizations of Weyl’s classical
equidistribution theorem: The first theorem applies to any polynomial map from
a locally compact amenable group to a compact abelian group. The second
theorem applies to any polynomial map from a countable group to a compact
abelian group.

##  1\. Equidistribution for polynomial maps

Weyl’s classical equidistribution theorem says that for any polynomial $P$
with real coefficients, the sequence $(\lfloor P(n)\rfloor)_{n=1}^{\infty}$ is
either uniformly distributed in $[0,1]$ or periodic. There are numerous
classical and more recent generalizations and refinements of Weyl’s
equidistribution theorem. For instance, Bergelson and Leibman obtained an
anlog of Weyl’s equidistribution theorem in finite characteristic [1].

The purpose of this note is to formulate and prove the following
generalization of Weyl’s classical theorem:

######  Theorem 1.1.

Let $P:\Gamma\to G$ be a polynomial map from a locally compact metrizable
amenable group $\Gamma$ to a compact metrizable abelian group $G$. Then:

  1. (1)

The set $\overline{P(\Gamma)}$, which is the closure of the image of $\Gamma$
under $P$, is a coset of a closed subgroup $H$ of $G$.

  2. (2)

The function $P:\Gamma\to G$ is well distributed with respect to the Haar
measure on $\overline{P(\Gamma)}$.

Our proof of Theorem 1.1 is based on a minor adaptation of the classical van
der Corput trick.

If we restrict ourselves to polynomial mappings from a countable discrete
group $\Gamma$ to a compact metrizable abelian group $G$, there is a
meaningful formulation of the result that does not require amenability of
$\Gamma$.

######  Theorem 1.2.

Let $P:\Gamma\to G$ be a polynomial map from a countable discrete group
$\Gamma$ to a compact metrizable abelian group $G$. Then:

  1. (1)

The $\Gamma$-orbit closure of $P$ in $\subseteq G^{\Gamma}$ is a coset of a
closed subgroup of $G^{\Gamma}$.

  2. (2)

There is a unique $\Gamma$-invariant measure on $\overline{\Gamma P}$, which
is Haar measure on $\overline{\Gamma P}$.

At first glance, it might seem surprising that equidistribution results can be
obtained under such general assumptions on the group $\Gamma$. This seeming
generality is somewhat deceiving: Liebman has shown that any polynomial map
taking values in a nilpotent group factors through a nilpotent quotient [3].
Liebman’s result allows us to easily derive Theorem 1.2 from Theorem 1.1,
despite its seemingly increased generality (no amenability assumption).

In the proof of both results we do use the fact that the “range” group $G$ is
abelian, since the proof involves Weyl’s equidistribution criterion via the
group of characters. It is natural to anticipate a nilpotent extension:
Leibman’s paper provides evidence that nilpotent groups are “the natural
framework” for results about polynomial maps. Green and Tao’s seminal work on
polynomial orbits on nilmanifolds [2] provides a plausible approach to such
generalization. In the case of degree $1$ polynomials, the analog of Theorem
1.1 is easy. We provide a short proof for this result in Section 5.

Finally, a remark about the compactness assumption of the “target” group $G$:
If $G$ is not assumed to be compact, already the statement that the closure of
the image of a polynomial map is a subgroup fails. Indeed, already in the
simplest example of the quadratic polynomial $\Gamma=G=\mathbb{Z}$ and
$P(n)=n^{2}$ we have $P(\mathbb{Z})=\\{0,1,2^{2},\ldots\\}$, which is not a
subgroup.

I thank Yaar Solomon and Matan Tal for their remarks on early versions of this
note, and Vitaly Bergelson for valuable comments, references, and
encouragement to make this writing publicly available.

##  2\. Definitions and conventions

Let us quickly introduce some notation and clarify definitions of various
terms in the statement of Theorem 1.1.

We use multiplicative notation for the group operation in a locally compact
second countable topological group $\Gamma$, whenever we do not assume
$\Gamma$ is commutative. We let $m_{\Gamma}$ denote left Haar measure on
$\Gamma$. For a compact group, we always assume that the Haar measure is
normalized to be a probability measure. Otherwise, when $\Gamma$ is not
compact, $m_{\Gamma}(\Gamma)=+\infty$ and we assume some (arbitrary) choice of
normalization for $m_{\Gamma}$.

For a compact abelian group $G$, we usually use additive notation for the
group operation. The one exception to this convention is the case where
$G=\mathbb{S}^{1}=\\{z\in\mathbb{C}^{*}:\leavevmode\nobreak\ |z|=1\\}$ is the
unit sphere in $\mathbf{C}$, in which case the group operation is the
multiplication of complex numbers, for which we use multiplicative notation.
Also, for a compact abelian group $G$ we let $\hat{G}$ denote the group of
characters of $G$, or the Pontryagin dual group, namely the continuous
homomorphism from $G$ to $\mathbb{S}^{1}$ .

If $G$ is a compact abelian group, $H<G$ is a closed subgroup and $C=v+H$ is a
coset of $H$ (where $v\in G$ is some element of the group), by “the Haar
measure on $C$” we mean the unique probability measure on $C$ which is
invariant to translation by elements of $H$. Equivalently, the Haar measure on
the coset $C$ is the pushforward of the Haar measure on the closed subgroup
$H$ via the map $x\mapsto x+v$ from $H$ to $C$.

In the following, $\Gamma$ will denote a locally compact second countable
topological group and $G$ a compact abelian group.

######  Definition 2.1.

For subset $F_{0},F\subset\Gamma$ and $\gamma\in\Gamma$ we denote:

$\partial_{\gamma}F=\gamma F\triangle F\mbox{ and
}\partial_{F_{0}}F=\bigcup_{\gamma\in F_{0}}\partial_{\gamma}F.$

For $\gamma\in\Gamma$ let $\sigma_{\gamma}:G^{\Gamma}\to G^{\Gamma}$ be given
by $\sigma_{\gamma}(\Phi)(x):=\Phi(\gamma^{-1}x)$.

A sequence $(F_{n})_{n=1}^{\infty}$ of measurable subset of $\Gamma$ having
finite non-zero Haar measure is called a _Følner sequence_ if for every
$\gamma\in\Gamma$ it holds that:

$\lim_{n\to\infty}\frac{m_{\Gamma}\left(\partial_{\gamma}F_{n}\right)}{m_{\Gamma}(F_{n})}=0.$

######  Definition 2.2.

For a function $P:\Gamma\to G$ and $\gamma\in\Gamma$, let
$\Delta_{\gamma}P:\Gamma\to G$ denote the discrete derivative of $P$ with
respect to $\gamma$:

$\Delta_{\gamma}P(x)=P(\gamma x)-P(x)\mbox{ for }x\in\Gamma.$

Note that for functions $\Phi:\Gamma\to\mathbb{S}^{1}$, since the group
operation is written as multiplication, we have

$\Delta_{\gamma}\Phi(x)=\Phi(\gamma x)\overline{\Phi(x)},$

where $\Phi(x)$ is the complex conjugate of $\Phi(x)$.

######  Definition 2.3.

A continuous function $P:\Gamma\to G$ is called a _Polynomial map of degree at
most $d-1$_ if

$\Delta_{\gamma_{d}}\ldots\Delta_{\gamma_{1}}P=0\leavevmode\nobreak\
\forall\gamma_{1},\ldots,\gamma_{d}\in\Gamma.$

A polynomial map of degree $0$ is a constant map from $\Gamma$ to $G$.

######  Definition 2.4.

Let $\Gamma$ be a locally compact amenable group.

For $\phi\in L^{\infty}(\Gamma)$ and a measurable set $F\Subset\Gamma$ with
$0<m_{\Gamma}(F)<\infty$, we denote

$A_{F}(\phi):=\frac{1}{m_{\Gamma}(F)}\int_{F}\phi(x)dm_{\Gamma}(x).$

We say that $f\in L^{\infty}(\Gamma)$ has _mean $z_{0}\in\mathbb{C}$_ if for
any Følner sequence $(F_{n})_{n=1}^{\infty}$ in $\Gamma$ the following holds:

$\lim_{n\to\infty}A_{F_{n}}(f)=z_{0}.$

Let $\phi:\Gamma\to X$ be a Borel measurable function from an amenable group
$\Gamma$ to a compact metrizable space $X$, and let $\mu$ be a Borel
probability measure on $X$. We say that $\phi$ is _equidistributed with
respect to $\mu$_ if for every continuous function $f:X\to\mathbb{C}$ the mean
of $f\circ\phi:\Gamma\to\mathbb{C}$ is $\int fd\mu$.

In other words, a function $\phi:\Gamma\to X$ is equidistributed with respect
to a probability measure $\mu$ on $X$ if for any Følner sequence
$(F_{n})_{n=1}^{\infty}$ in $\Gamma$ the measures
$\frac{1}{m_{\Gamma}(F_{n})}\int_{F_{n}}\delta_{\phi(\gamma)}dm_{\gamma}(\gamma)$
converge weak-$*$ to $\mu$.

##  3\. Proof of the equidistribution theorem

The proof of Theorem 1.1 is based on two lemmas.

The first says that if a function into $\mathbb{S}^{1}$ has some constant but
non-trivial directional derivative, then that function must have zero mean:

######  Lemma 3.1.

Let $\Phi:\Gamma\to\mathbb{S}^{1}$ be a measurable function. If there exists
$\gamma\in\Gamma$ and $z_{0}\in\mathbb{S}^{1}\setminus\\{1\\}$ such that
$\Delta_{\gamma}\Phi(x)=z_{0}$ for all $x\in\Gamma$ then $\Phi$ has mean zero.

###### Proof.

Suppose $z_{0}\in\mathbb{S}^{1}\setminus\\{1\\}$ and that
$\Delta_{\gamma}\Phi(x)=z_{0}$ for all $x\in\Gamma$. By definition,

$\Phi(\gamma x)=\Delta_{\gamma}\Phi(x)\Phi(x)=z_{0}\Phi(x).$

So for every $k\in\mathbb{Z}$ $\Phi(\gamma^{k}x)=z_{0}^{k}\Phi(x)$. Because
$|\Phi(x)|\leq 1$, for any compact set $F\Subset\Gamma$ we have

$\left|\int_{F}\Phi(x)dm_{\Gamma}(x)-\int_{F}\Phi(\gamma
x)dm_{\Gamma}(x)\right|\leq m_{\Gamma}(\partial_{\gamma}(F)).$

Substituting $\Phi(\gamma x)=z_{0}\Phi(x)$ it follows that

$\left|\int_{F}\Phi(x)dm_{\Gamma}(x)-\int_{F}z_{0}\Phi(x)dm_{\Gamma}(x)\right|<m_{\Gamma}\left(\partial_{\gamma}(F)\right).$

Using $z_{0}\neq 1$, we have:

$\left|\int_{F}\Phi(x)dm_{\Gamma}(x)\right|<\frac{1}{|1-z_{0}|}m_{\Gamma}\left(\partial_{\gamma}(F)\right).$

Let $(F_{n})_{n=1}^{\infty}$ be a Følner sequence in $\Gamma$. For each
$n\in\mathbb{N}$ we have:

$\left|\frac{1}{m_{\Gamma}(F_{n})}\int_{F_{n}}\Phi(x)dm_{\Gamma}(x)\right|<\frac{1}{|1-z_{0}|}\frac{m_{\Gamma}\left(\partial_{\gamma}(F_{n})\right)}{m_{\Gamma}(F_{n})}.$

Taking $n\to\infty$ we conclude that $\Phi$ has mean zero.

∎

The second lemma is a version of the well-known van der Corput Lemma :

######  Lemma 3.2 (The van der Corput Lemma for maps from amenable groups to
compact abelian groups).

Let $\Phi:\Gamma\to G$ be a function such that $\Delta_{\gamma}\Phi:\Gamma\to
G$ is equidistributed with respect to the Haar measure on $G$ for any
$\gamma\in\Gamma\setminus\\{1_{\Gamma}\\}$. Then $\Phi$ is equidistributed
with respect to the Haar measure on $G$.

The proof of Lemma 3.2 is based in the following version of the van der Corput
inequality:

######  Lemma 3.3 (The Van der Corpout inequality for complex valued functions
on amenable groups).

Let $\Gamma$ be a discrete group, and let $F,F_{0}\subset\Gamma$ be Borel
subsets of $\Gamma$ having finite positive Haar measure. Let
$\phi:\Gamma\to\mathbb{C}$ be a function with $|\phi(g)|\leq 1$ for all
$g\in\Gamma$. Then

$|A_{F}(\phi)|\leq\frac{1}{m_{\Gamma}(F_{0})}\sqrt{\int_{F_{0}}\int_{F_{0}}A_{F}\left(\sigma_{\gamma_{2}^{-1}}(\Delta_{\gamma_{1}^{-1}\gamma_{2}}\phi)\right)dm_{\Gamma}(\gamma_{1})dm_{\Gamma}(\gamma_{2})}+\frac{m_{\Gamma}(\partial_{F_{0}}F)}{m_{\Gamma}(F)}$
(1)

The proof here is essentially an imitation of the classical one. The
presentation here is closely inspired by the post on Terry Tao’s blog “The van
der Corput trick, and equidistribution on nilmanifolds” from 2008:

###### Proof.

Observe that for any $f:\Gamma\to\mathbb{C}$ with $|f|\leq 1$ and any
$F_{0},F\Subset\Gamma$, $\gamma\in F_{0}$ we have

$\left|A_{F}(f)-A_{F}(\sigma_{\gamma}f)\right|\leq\frac{m_{\Gamma}(\partial_{F_{0}}F)}{m_{\Gamma}(F)}.$
(2)

Taking $f=\phi$, averaging over $\gamma\in F_{0}$ and using the triangle
inequality we obtain

$\left|A_{F}(\phi)-A_{F}\left(\frac{1}{m_{\Gamma}(F_{0})}\int_{F_{0}}\left(\sigma_{\gamma}\phi\right)dm_{\Gamma}(\gamma)\right)\right|\leq\frac{m_{\Gamma}(\partial_{F_{0}}F)}{m_{\Gamma}(F)}.$

Applying the triangle inequality again and the Cauchy-Schwarz inequality we
get:

$\left|A_{F}(\phi)\right|\leq\left[A_{F}\left(\left|\frac{1}{m_{\Gamma}(F_{0})}\int_{F_{0}}\sigma_{\gamma}\phi
dm_{\Gamma}(\gamma)\right|^{2}\right)\right]^{\frac{1}{2}}+\frac{m_{\Gamma}(\partial_{F_{0}}F)}{m_{\Gamma}(F)}.$

Expending

$\left|\frac{1}{m_{\Gamma}(F_{0})}\int_{F_{0}}\sigma_{\gamma}\phi
dm_{\Gamma}(\gamma)\right|^{2}=\frac{1}{m_{\Gamma}(F_{0})^{2}}\left(\int_{F_{0}}\int_{F_{0}}\sigma_{\gamma_{1}}\phi\overline{\sigma_{\gamma_{2}}\phi}dm_{\Gamma}(\gamma_{1})dm_{\Gamma}(\gamma_{2})\right),$

Using that
$\sigma_{\gamma_{1}}\phi\overline{\sigma_{\gamma_{2}}\phi}=\sigma_{\gamma_{2}^{-1}}(\Delta_{\gamma_{1}^{-1}\gamma_{2}}\phi)$
we get:

$|A_{F}(\phi)|\leq\frac{1}{m_{\Gamma}(F_{0})}\sqrt{\int_{F_{0}}\int_{F_{0}}A_{F}\left(\sigma_{\gamma_{2}^{-1}}(\Delta_{\gamma_{1}^{-1}\gamma_{2}}\phi)\right)dm_{\Gamma}(\gamma_{1})dm_{\Gamma}(\gamma_{2})}+\frac{m_{\Gamma}(\partial_{F_{0}}F)}{m_{\Gamma}(F)}$

∎

One final ingredient that we need is the well known criterion of Weyl for
equidistribution:

######  Lemma 3.4 (Weyl’s equidistribution criterion).

A function measurable function $\Phi:\Gamma\to G$ from a locally compact
amenable group to a compact metrizable abelian group $G$ is equidistributed if
and only if $\chi\circ\Phi$ has mean zero for every non-trivial character
$\chi\in\hat{G}$.

Indeed, this follows since any continuous function $f:G\to\mathbb{C}$ can be
uniformly approximated by characters.

###### Proof of Lemma 3.2.

Let $\Phi:\Gamma\to G$ be as in the statement. By Weyl’s equidistribution
criterion, it suffices to prove that $\phi:=\chi\circ\Phi$ has mean zero for
any non-trivial character $\chi\in\hat{G}$. Since
$\Delta_{\gamma}(\chi\circ\Phi)=\chi\circ\Delta_{\gamma}\Phi$, the result
follows from Equation 1 ∎

######  Definition 3.5.

Given $\Phi:\Gamma\to G$, let

$\ker(\Phi)=\left\\{\gamma\in\Gamma\leavevmode\nobreak\ :\leavevmode\nobreak\
\Delta_{\gamma}\Phi(x)=0\leavevmode\nobreak\ \forall x\in\Gamma\right\\}.$

The following easy lemma justifies the name $\ker(\Phi)$:

######  Lemma 3.6.

For any continuous map $\Phi\in\Gamma\to G$, we have that $\ker(\Phi)$ is a
closed normal subgroup of $\Gamma$. The function $\Phi$ factors through a map
$\overline{\Phi}$ from the quotient of $\Gamma/\ker(\Phi)$.

###### Proof.

Let $X_{\Phi}$ denote the closure of
$\\{\sigma_{\gamma}(\Phi)):\leavevmode\nobreak\ \gamma\in\Gamma\\}$ in the
space of continuous functions from $\Gamma$ to $G$, with the topology of
uniform convergence on compact sets. The map $\gamma\mapsto\sigma_{\gamma}$ is
a continuous homomorphism from $\Gamma$ to the group of homeomorphisms of
$X_{\Phi}$, and $\ker(\Phi)$ is exactly the kernal of this homomorphism. ∎

###### Proof of Theorem 1.1.

We prove the theorem by induction on the degree $d$ of the polynomial map $P$.
The basis for the induction is the trivial case $d=0$: Indeed, if $d=0$, then
$P$ is constant, and $P(\Gamma)$ is a singleton, which is a coset of the
trivial group, and $P$ is obviously well distributed on this (trivial) coset.

Now suppose $P:\Gamma\to G$ is a polynomial map of degree $d\geq 1$. Let
$G_{0}$ denote the closed group generated by all elements of the form
$\Delta_{\gamma}P(x)$ where $\gamma,x\in\Gamma$. Clearly $P$ takes values in
$P(1_{\Gamma})+G_{0}$. By substracting a constant from $P$ we can assume
without loss of generality that $P(1_{\Gamma})=0$, and under this assumption
our goal is to prove that $P$ is equidistributed on $G_{0}$. With the above
reductions in mind, our goal is to show that $P$ is well distributed on
$G_{0}$. By Weyl’s equidistribution criterion it suffices to show that for any
character $\chi$ that does not vanish on $G_{0}$, $\chi\circ
P:\Gamma\to\mathbb{C}$ has mean zero. Clearly, a character $\chi\in\hat{G}$
vanishes on $G_{0}$ if and only if $\chi(\Delta_{\gamma}P(x))=0$ for every
$\gamma,x\in\Gamma$. Let $\chi$ be a character that does not vanish on
$G_{0}$. For every $\gamma\in\Gamma$,
$\chi\circ\Delta_{\gamma}P=\Delta_{\gamma}(\chi\circ P)$ is a polynomial map
of degree less than $d$. Thus, by the induction hypothesis,
$\chi\circ\Delta_{\gamma}P$ is equidistributed on the closure of
$\chi\circ\Delta_{\gamma}P(\Gamma)$, which is a coset of closed subgroup of
$\mathbb{S}^{1}$. If $\chi\circ\Delta_{\gamma}P$ is equidistributed on a coset
of a _non-trivial_ subgroup of $\mathbb{S}^{1}$, it has mean zero. Thus, by
Lemma 3.2 we are done unless there exists
$\gamma\in\Gamma\setminus\\{1_{\Gamma}\\}$ for which
$\chi\circ\Delta_{\gamma}P$ is supported (and equidistributed) on a coset of
the trivial group, namely, $\Delta_{\gamma}(\chi\circ P)$ is constant. By
Lemma 3.6 we can replace $\Gamma$ with $\Gamma/\ker(\chi\circ P)$, thus
assuming $\ker(\chi\circ P)$ is the trivial subgroup. So suppose there exists
$\gamma\in\Gamma\setminus\\{1_{\Gamma}\\}$ for which
$\chi\circ\Delta_{\gamma}P$ is identically equal to a constant $z_{0}\neq 1$
$\chi\circ P$. By assumption $\gamma\neq 1_{\Gamma}$ and $\ker(\chi\circ
P)\\{1_{\Gamma}\\}$, so $z_{0}\neq 1$. It follows directly from Lemma 3.1 that
$\chi\circ P$ has mean zero. ∎

##  4\. Polynomials into locally compact groups with a compact image

Leibman [3, Proposition 1.21] has shown that a polynomial map from a countable
group $\Gamma_{1}$ to a countable torsion free group $\Gamma_{2}$ that takes
finitely many values must be constant. The following is a direct
generalization of Leibman’s observation to the locally-compact setting:

######  Proposition 4.1.

Let $P:\Gamma_{1}\to\Gamma_{2}$ be a polynomial map between locally compact
groups $\Gamma_{1}$ and $\Gamma_{2}$. Suppose that the only subgroup of
$\Gamma_{2}$ with compact closure is the trivial group. If
$\overline{P(\Gamma_{1})}$ is a compact subset of $\Gamma_{2}$ then $P$ is
constant.

###### Proof.

The short proof is nothing but a a direct adaptation of Leibman’s proof to
this more general setting: The proof is by induction on the degree of $P$. If
$\deg(P)=0$, then $P$ is constant, and the claim is trivial. If $\deg(P)=1$,
then $P$ is of the form $P(\gamma)=h(\gamma)\gamma_{0}$, for some continuous
homomorphism $h:\Gamma_{1}\to\Gamma_{2}$ and $\gamma_{0}\in\Gamma$. So
assuming that $\overline{P(\Gamma_{1})}$ is compact and that hat the only
subgroup of $\Gamma_{2}$ with compact closure is the trivial group, we
conclude in this case that $\overline{P(\Gamma_{1})}$ is a coset of the
trivial group, in other words that $P$ is constant. Now assume the conclusion
of the lemma holds for polynomial maps of degree at most $d$ for some $d\geq
1$. Suppose $\deg(P)=d+1$ For any $\gamma_{1}\in\Gamma_{1}$, we have that
$\Delta_{\gamma_{1}}P(\Gamma_{1})\subseteq P(\Gamma_{1})(P\Gamma_{1})^{-1}$.
Hence, If $P(\Gamma_{1})$ has compact closure so does
$\Delta_{\gamma_{1}}P(\Gamma_{1})$. By the induction hypothesis, since
$\deg(\delta_{\gamma_{1}}P)\leq d$, we have that
$\Delta_{\gamma_{1}}P(\Gamma_{1})$ must in this case be a singleton, under the
assumption that the only subgroup of $\Gamma_{2}$ with compact closure is the
trivial group. This means that $\deg(P)=1$, a contradiction. ∎

Proposition 4.1 shows for instance that any bounded polynomial map from a
locally compact group $\Gamma$ into $\mathbb{R}$ or $\mathbb{C}$ must be
constant.

##  5\. Equidistribution of homomorphisms into non-abelian compact groups

######  Theorem 5.1.

Let $\Phi:\Gamma\to G$ be a continuous homomorphism from a locally compact
metrizable amenable group $\Gamma$ to a compact metrizable group $G$. Then
$G_{\Phi}:=\overline{\Phi(\Gamma)}$, the closure of the image of $\Gamma$
under $\Phi$, is a closed subgroup of $G$, and $\Phi$ is equidistributed with
respect to Haar measure on $G_{\Phi}$.

###### Proof.

The image of a group under a homomorphism is always a group. Let
$(F_{n})_{n=1}^{\infty}$ be a Følner sequence in $\Gamma$, and let $\mu$ be a
probability measure on $X$ which is an accumulation point of
$\mu_{n}:=\frac{1}{m_{\Gamma}(F_{n})}\int_{F_{n}}\delta_{F(\gamma)}dm_{\gamma}(\gamma)$.
Our goal is to prove that $\mu$ is equal to Haar measure on $G_{\Phi}$. Since
$\mu_{n}(G_{\Phi})=1$, it is clear that the topological support of the measure
$\mu$ is equal to $G_{\Phi}$. It thus suffices to prove that $\mu$ is
invariant with respect to translation by elements of $G_{\Phi}$. Let $f\in
C(G)$ be a continuous function on $G$. We need to prove that

$\int f(g)\mu(g)=\int f(gh)\mu(g)\mbox{ for all }g\in G_{F},$

Since the function $h\mapsto\int f(gh)d\mu(g)$ from $G$ to $\mathbb{C}$ is
continuous, it suffices to prove the above for a dense subset of $G_{\Phi}$.
Specifically, it is enough to show that

$\int f(g)d\mu(g)=\int f(g\Phi(\gamma))\mu(g)\mbox{ for all }\gamma\in\Gamma.$

To prove this, it suffices to prove that

$\lim_{n\to\infty}\left|\int f(g)d\mu_{n}(g)-\int
f(g(\Phi(\gamma))d\mu_{n}(g)\right|=0.$

Indeed,

$\left|\int f(g)d\mu_{n}(g)-\int
f(g(F(\gamma))d\mu_{n}(g)\right|\leq\|f\|_{C(G)}\cdot\frac{m_{\Gamma}(F_{n}\triangle
F_{n}\Phi(\gamma))}{m_{\Gamma}(F_{n})}\to 0\mbox{ as }n\to\infty.$

∎

##  6\. Unique ergodicity of orbit closures of polynomial maps

In this section we prove Theorem 1.2 and recall the relevant definitions.

In what follows, $\Gamma$ will be a discrete countable group and $G$ a compact
abelian group. The space $G^{\Gamma}$ of functions from $\Gamma$ to $G$ is
equipped with the product topology, which makes it into a compact topological
space. It is furthermore a compact abelian group, with respect to the
pointwise addition in $G$. The group $\Gamma$ acts on $G^{\Gamma}$ by
homemorphisms (the shift action). This action is furthermore an _algebraic
action_ : Each element of $\Gamma$ acts as continuous automorphism of the
group $G^{\Gamma}$. A polynomial map $P:\Gamma\to G$ is by definition an
element of $G^{\Gamma}$. We denote by $\overline{\Gamma P}$ the closure in
$G^{\Gamma}$ of the orbit of $P$ under the action of $\Gamma$.

Any function $\Phi:\Gamma\to G$ naturally defines a function
$\Phi^{\Gamma}:\Gamma\to G^{\Gamma}$ by

$\Phi^{\Gamma}(\gamma)=\sigma_{\gamma}(\Phi)\leavevmode\nobreak\
\gamma\in\Gamma.$

Note that for any $\Phi\in G^{\Gamma}$

$\overline{\Gamma\Phi}=\overline{\Phi^{\Gamma}(\Gamma)}=\overline{\left\\{\sigma_{\gamma}(\Phi):\leavevmode\nobreak\
\gamma\in\Gamma\right\\}}.$

The following lemma is a straightforward but crucial observation:

######  Lemma 6.1.

If $P:\Gamma\to G$ is a polynomial map of degree $d$ then
$P^{\Gamma}:\Gamma\to G^{\Gamma}$ is a polynomial map of degree $d$.

The proof of Theorem 1.2 can be completed as follows:

###### Proof of Theorem 1.2.

Let $P:\Gamma\to G$ be a polynomial map, where $\Gamma$ is a discrete
countable group and $G$ is a compact abelian group.

By Leibman’s result [2, Proposition 3.21] there exists a nilpotent group
$\Gamma_{0}$, a homomorphism $h:\Gamma\to\Gamma_{0}$ and a polynomial map
$P_{0}:\Gamma_{0}\to G$ such that $P=P_{0}\circ h$. By Lemma 6.1 the map
$P_{0}^{\Gamma_{0}}:\Gamma_{0}\to G^{\Gamma_{0}}$ is also a polynomial map.
Since nilpotent groups are amenable, It follows from Theorem 1.1 that
$\overline{P_{0}^{\Gamma_{0}}(\Gamma_{0})}$ is a coset of a closed subgroup of
$G^{\Gamma_{0}}$, and that $P_{0}^{\Gamma_{0}}$ is well distributed with
respect to Haar measure on this coset. By subtracting a constant we can assume
without loss of generality that $P(1_{\Gamma})=0$. Under this assumption
$P(1_{\Gamma})=0$ and the coset $\overline{P_{0}^{\Gamma_{0}}(\Gamma_{0})}$ is
actually a closed subgroup. But from the definition of $P_{0}^{\Gamma_{0}}$ it
follows directly that $\overline{P_{0}^{\Gamma_{0}}(\Gamma_{0})}$ is precisely
equal to $\overline{\Gamma_{0}P_{0}}$. The orbit closure
$\overline{\Gamma_{0}P_{0}}$ is clearly invariant under the action of
$\Gamma_{0}$, and since this is a subgroup. By uniquness of the Haar measure,
the Haar measure on the subgroup $\overline{\Gamma_{0}P_{0}}$ is indeed
$\Gamma_{0}$-invariant. Using the erogdic theorem (for actions of nilpotent
group), for any $\Gamma_{0}$ invariant measure $\mu$ on
$\overline{\Gamma_{0}P_{0}}$ one can find a Følner sequence
$(F_{n})_{n=1}^{\infty}$ in $\Gamma_{0}$ such that

$\lim_{n\to\infty}\frac{1}{m_{\Gamma_{0}}(F_{n})}\sum_{\gamma_{0}\in
F_{0}}\delta_{P_{0}^{\Gamma_{0}}(\gamma)}=\mu,$

where the convergence is weak-*. But the fact that $P_{0}^{\Gamma_{0}}$ is
equidistributed with respect to Haar measure on
$\overline{P_{0}^{\Gamma_{0}}(\Gamma_{0})}$ implies that any such $\mu$ is
equal to Haar measure on $\overline{P_{0}^{\Gamma_{0}}(\Gamma_{0})}$.

The last step of the proof is to “lift” everything from $\Gamma_{0}$ to
$\Gamma$ via the homomorphism $h$. ∎

## References

  * [1] V. Bergelson and A. Leibman, _A Weyl-type equidistribution theorem in finite characteristic_ , Adv. Math. 289 (2016), 928–950. MR 3439703
  * [2] Ben Green and Terence Tao, _The quantitative behaviour of polynomial orbits on nilmanifolds_ , Ann. of Math. (2) 175 (2012), no. 2, 465–540. MR 2877065
  * [3] A. Leibman, _Polynomial mappings of groups_ , Israel J. Math. 129 (2002), 29–60. MR 1910931