# Universal lower bounds for the discrepancies of actions of a locally compact
group

Antoine Pinochet Lobos  and  Christophe Pittet

###### Abstract.

We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps
of averaging operators) of measure-preserving actions of a locally compact
group on probability spaces. For example, a locally compact Hausdorff
unimodular group $G$, acting continuously, by measure-preserving
transformations, on a compact atomless probability space $(X,\nu)$, with an
orbit $Gx_{0}$ of measure zero, contained in the support of $\nu$, and with
compact stabilizer (i.e. $G_{x_{0}}$ is compact) has the following property:
any finite positive regular Borel measure $\mu$ on $G$ satisfies

$\|\pi_{0}(\mu)\|_{2\to 2}\geq\|\lambda_{G}(\mu)\|_{2\to 2},$

where $\pi_{0}$ denotes the Koopman representation of $G$, defined by the
given action, and $\lambda_{G}$ denotes the left-regular representation of
$G$. The lower bounds we prove generalize the universal lower bounds for the
discrepancies of measure-preserving actions of a discrete group. Many examples
show that the generalization from discrete groups to locally compact groups
requires some additional hypothesis on the action (we detail some examples of
actions of amenable groups with a spectral gap, due to Margulis). Well-known
examples and results of Kazhdan and Zimmer show that the discrepancies of some
actions of Lie groups on homogeneous spaces match exactly the universal lower
bounds we prove.

###### Key words and phrases:

Atomless measure, convolution operator, discrepancy, Koopman representation,
locally compact group, measure-preserving action, regular representation,
spectral gap, unitary representation, von Neumann ergodic theorem, weak
containment.

###### 2020 Mathematics Subject Classification:

Primary: 37A15 ; Secondary: 37A30

I2M, CNRS UMR7373, Université d’Aix-Marseille, a.p.lobos _at_ outlook.com

I2M, CNRS UMR7373, Université d’Aix-Marseille et Section de Mathématiques,
Faculté des Sciences, Université de Genève, pittet _at_ math.cnrs.fr
The authors acknowledge support of the FNS grant 200020-200400.

###### Contents

  1. 1 Introduction
  2. 2 Outline of the paper
  3. 3 Preliminaries
  4. 4 Statement of the results
  5. 5 Interpretation of the technical condition in Proposition 2
  6. 6 Proofs
    1. 6.1 Proof of Proposition 2
    2. 6.2 Nets and volume estimates
    3. 6.3 Proof of Theorem 1 and Corollary 1
  7. 7 Examples and counterexamples
    1. 7.1 Margulis’ counterexample
    2. 7.2 An action on a finite-volume homogeneous space with optimal discrepancy

##  1\. Introduction

The discrepancies of a measure-preserving group action are related to the
possible rates of convergence in ergodic von Neumann type theorems. They
measure the sizes of spectral gaps of averaging operators. See Definition 3.1
and Proposition 1 below. The notion of discrepancy can be interpreted as a
measure of the ergodicity of the action [ShaTata, Thm 4.2].

Isometric group actions, on the $2$-dimensional sphere, of free groups with
the smallest possible discrepancies (equivalently, with the fastest possible
rate of convergence) have been constructed by Lubotzky, Phillips, and Sarnak.
See [LPS] and [PinPitExactRate]. For discrete groups, there exist universal
lower bounds for the discrepancies of measure-preserving actions on atomless
probability spaces, which can be expressed with the help of the regular
representation. In other words, there are lower bounds for the discrepancies,
which are valid for any measure-preserving action of the group on an atomless
probability space, and these universal lower bounds, which are inherent to the
structure of the group, can be extracted from the regular representation of
the group. The above mentioned actions considered by Lubotzky, Phillips, and
Sarnak, have discrepancies matching exactly these universal lower bounds. See
[PinPitExactRate].

A remarkable and well-known connection between dynamical systems and the
theory of unitary representations is that discrepancies of locally compact
group actions are equal to the norms of operators associated to the
corresponding Koopman representations (see Proposition 1 below for the precise
statement we will use). In other words, the discrepancies we consider (as
initially defined in [LPS, 0.3 and 0.3’]) measure the sizes of the spectral
gaps of Koopman operators. These norms, equivalently these spectral gaps, have
been studied by many authors (see for example [ShaTata] [FurSha], [BekGui],
[GorNev], [GorNevBook]). Spectral transfer, in the sense of Nevo [Nev], is a
powerful method to estimate these norms. Roughly speaking, the idea is to
apply different unitary representations, let’s say $\pi$ and $\lambda$, of a
locally compact group $G$, to a given probability measure $\mu$ on $G$, and to
compare the operator norms of the operators $\pi(\mu)$ and $\lambda(\mu)$
acting on different Hilbert spaces. See also [GorNev, 4. Spectral gaps and
ergodic theorems]. For example, one of the main point in the work of Lubotzky,
Phillips, Sarnak, cited above, is the equality between the spectrum of
$\pi_{0}(\mu_{n})$ and the one of $\lambda_{G}(\mu_{n})$, where $\pi_{0}$ is a
specific (defined by arithmetic conditions) Koopman representation on the
$2$-dimensional sphere of a finitely generated free group $G$, where
$\lambda_{G}$ is the left regular representation of $G$, and where $\mu_{n}$
is the uniform measure on the sphere of radius $n$ with center the identity
$e\in G$ with respect to the word metric defined by the canonical generating
set of $G$. When $n=1$, the equality of the spectra is proved with the help of
Deligne’s solution to the Weil conjecture (see [LPS2, page 419] and [Lub]),
the cases with $n>1$ follow from the case $n=1$ and from the spectral theorem
applied to Hecke elements. Exact computations can be performed with the help
of the Harish-Chandra function of the natural boundary representation of the
free group (see [PinPitExactRate]). In [Nev, Theorem 1], Nevo applies the
spectral transfer method to any Kazhdan connected semi-simple Lie group $G$
with finite center and without compact factor. Relying on estimates of some
coefficients of unitary representations of $G$, due to Cowling [Cow], Nevo
shows that there is an even integer $n(G)$, such that the Koopman
representation $\pi_{0}$ of $G$, defined by any measure-preserving ergodic
action of $G$ on a probability space, satisfies, for any Borel probability
measure $\mu$ on $G$, $\|\pi_{0}(\mu)\|_{2\to 2}\leq\|\lambda_{G}(\mu)\|_{2\to
2}^{1/n(G)}$. In some special cases, it is known (see [Nev, 4.1] and
Proposition 5 below) that the upper bound on $\|\pi_{0}(\mu)\|$ (we drop the
$2\to 2$ indexes) can be strengthened to
$\|\pi_{0}(\mu)\|\leq\|\lambda_{G}(\mu)\|$.

Most of the results on discrepancies are upper bounds (one tries to get the
fastest possible rate of convergence, by proving $\|\pi_{0}(\mu)\|<1$ is as
small as possible, equivalently by proving the size of the spectral gap is as
large as possible). But lower bounds have also been studied, for example in
[Sev, Théorème 3.3], [Clo, Theorem 2], [Pis, Theorem 4], [LPS, Theorem 1.3].
All these lower bounds concern discrete groups. Proposition 2 below
generalizes all these lower bounds, and applies to a general locally compact
Hausdorff group $G$. Its conclusion is the lower bound

(1) $\displaystyle\|\pi_{0}(\mu)\|\geq\|\lambda_{G}(\mu)\|,$

for any positive bounded Borel regular measure $\mu$ on $G$, where $\pi_{0}$
is the Koopman representation defined by a measure-preserving $G$-action on an
atomless probability space, satisfying technical hypothesis we will comment
on. The proof of Proposition 2 is inspired from the proof of [PinPitExactRate,
Theorem 3.6], which in turn is based on the idea of the proof of [Sev,
Théorème 3.3]. In contrast with the case of discrete groups, Inequality (1)
can’t be true for general measure preserving actions of general locally
compact Hausdorff groups on an atomless probability space: there are many
examples of amenable Lie groups acting by measure preserving transformations
on atomless probability spaces which have a spectral gap (see Section 7.1).
Condition (3) of Proposition 2 (which is trivially fulfilled in the case $G$
is discrete) is an additional assumption on the action which makes possible to
handle the general case of a locally compact Hausdorff group $G$.
Heuristically, Condition (3) guarantees that Følner quotients of partial
orbits growth sub-linearly (see Subsection 5 below for an example and some
pictures). Condition (3) potentially excludes some interesting actions, but we
show in (the proof of) Theorem 1 that it is fulfilled in many natural
situations.

Shalom, and independently, Dudko and Grigorchuk, have proven that for any
discrete group acting by measure-preserving transformations on any atomless
probability space, the associated Koopman representation $\pi_{0}$ (i.e. the
one on square integrable functions of zero integral, which is the one we study
in this paper) weakly contains the quasi-regular representation associated
with almost every point stabilizer; see [DudGri, Proposition 7] (where the
atomless probability space is assumed to be standard) and [ShaTata] (where the
atomless probability space is not assumed to be standard but the result is
stated without proof). This weak containment result implies Inequality (1) for
all positive finite regular Borel measure $\mu$: one applies the
characterization of weak containment explained in [BekDelaHa, Remark 8.B.6 (2)
page 244], and an inequality due to Shalom [ShaAnnals, Lemma 2.3] (which
requires the measure $\mu$ to be positive). The same conclusion, namely
Inequality (1) for all positive finite regular Borel measure $\mu$, is
obtained directly by applying Proposition 2. In the case there exists a
trivial stabilizer, Inequality (1) is true for any bounded Borel regular
measure $\mu$, not only for positive $\mu$; equivalently
$\pi_{0}\succ\lambda_{G}$.

In the case $G$ is totally discontinuous and second countable, Caprace,
Kalantar, and Monod, prove an interesting weak containment result, for some
measurable actions, preserving _the class_ of the measure, on a standard
probability space (see [CapraceKalantarMonod, Theorem F]) but it seems that no
generalization of the weak containment result of Shalom and Dudko-Grigorchuk
mentioned above is known for locally compact Hausdorff groups. Notice that
[CapraceKalantarMonod, Theorem F] says nothing about Inequality (1): take $G$
a compact group with Haar measure $\mu_{G}$, for example $G=\mathbb{Z}_{p}$
the $p$-adic integers, and $\pi_{0}$ the Koopman representation on
$L^{2}_{0}(G,\mu_{G})$, then $\pi_{0}(\mu_{G})=0$, whereas the amenability of
$G$ implies $\|\lambda_{G}(\mu_{G})\|=\mu_{G}(G)$.

In the proof of Theorem 1, we use a technic from geometric group theory
(volume approximation in a locally compact group based on nets relative to a
word metric) to show that if $G$ is locally compact Hausdorff unimodular and
acts continuously and freely (more generally with compact stabilizers) with an
orbit of measure zero (contained in the support of the measure), then all the
hypothesis of Proposition 2 are fulfilled, hence Inequality (1) holds in this
case (the existence of a orbit with zero measure is a necessary condition as
many examples show: see Section 7.1). We do not know if the compact stabilizer
hypothesis and/or the unimodularity hypothesis in Theorem 1 can be removed
(but the proof we give, using nets, needs both hypothesis).

An application of Theorem 1 is the following. Suppose $H$ is a closed
unimodular subgroup of a Lie group $G$ and $\Gamma$ is a lattice in $G$ such
that $\Gamma\cap H$ is finite. If the dimension of $H$ is smaller than the one
of $G$, then the Koopman representation associated to the natural action of
$H$ on $G/\Gamma$ satisfies Inequality (1) for any positive bounded Borel
regular measure on $H$. See Corollary 1. In Proposition 5 we give examples of
actions of Lie groups where the lower bounds brought by Theorem 1 and
Corollary 1 match exactly upper bounds deduced from well known results of
Kazhdan and Zimmer.

##  2\. Outline of the paper

In Section 3 we fix notation and briefly recall well-known links between
discrepancies, operator norms, and Koopman representations. In Section 4 we
state the main results of the paper. In Section 5 we illustrate Condition (3)
from Proposition 2 with an example and two pictures. Section 6 is devoted to
the proofs of the results stated in Section 4. In Section 7.1, we recall a
result of Margulis implying the existence of plenty of natural actions of
amenable groups with a spectral gap. Finally, we describe in Section 7.2 an
action of $\mathbb{R}^{2}\rtimes\operatorname{SL}_{2}(\mathbb{R})$ on a
finite-volume homogeneous space, where the inequality of Corollary 1 is sharp.

_Acknowledgements._ We warmly thank Pierre-Emmanuel Caprace who was at the
origin of the questions that we address in this paper; we are grateful to him
for helpful discussions and remarks.

##  3\. Preliminaries

In this paper, $G$ denotes a locally compact Hausdorff topological group and
$\mu_{G}$ is a left Haar measure on $G$.

######  Definition 3.1 (Discrepancy).

Let $(X,\mathcal{T},\nu)$ be a probability space, and $G\curvearrowright X$ a
measurable action which preserves $\nu$. Let $\mu$ be a Borel probability
measure on $G$. We call the number

$\delta(\mu):=\sup_{\begin{subarray}{c}\phi\in L^{2}(X,\nu)\\\
\|\phi\|_{2}=1\end{subarray}}\left\|x\mapsto\int_{G}\phi\left(g^{-1}x\right)\mathop{}\\!\mathrm{d}\mu-\int_{X}\phi\mathop{}\\!\mathrm{d}\nu\right\|_{2}.$

the discrepancy of $\mu$. Similarly, for any continuous, non-negative,
compactly-supported function $f$ on $G$ of integral $1$, we denote by
$\delta(f)$ the discrepancy of the probability measure with density $f$ with
respect to $\mu_{G}$.

We denote

$\begin{array}[]{rcl}\lambda_{G}:G&\rightarrow&U(L^{2}(G,\mu_{G}))\\\
g&\mapsto&\displaystyle\phi\mapsto\left(h\mapsto\phi(g^{-1}h)\right)\\\
\end{array}$

the left regular representation of $G$.

######  Definition 3.2 (Koopman representation).

Consider an action of $G$ on a measure space $(X,\nu)$ by measurable
transformations preserving the measure.

The Koopman representation associated to this action is defined by

$\begin{array}[]{rcl}\pi:G&\rightarrow&U(L^{2}(X,\nu))\\\
g&\mapsto&(\phi\mapsto(x\mapsto\phi(g^{-1}x)))\\\ \end{array}$

######  Remark 1.

  1. (1)

The left regular representation is the Koopman representation for the action
of the locally compact Hausdorff group $G$ endowed with a left Haar measure
$\mu_{G}$ by left multiplication on itself.

  2. (2)

For every $g\in G$, the operator $\pi(g)$ defined above is unitary and $\pi$
is a homomorphism. Moreover, if $G$ is $\sigma$-compact, locally compact
Hausdorff, if $\nu$ is $\sigma$-finite and $L^{2}(X,\nu)$ is separable, then
the Koopman representation $\pi$ is a unitary representation (that is, it is
continuous if we endow $U(L^{2}(X,\nu))$ with the strong operator topology;
see [BHV, Proposition A.6.1]). When we consider a Koopman representation, or
any unitary representation, it is always implicitly assumed that it is
strongly continuous.

  3. (3)

If $\nu$ is a finite measure, then $\textbf{1}_{X}$ is a fixed vector, and its
orthogonal complement is a sub-representation. This subspace is the subset of
all (representatives of) functions with zero integral and we denote it by
$L^{2}_{0}(X,\nu)$. We denote the associated unitary sub-representation

$\pi_{0}:G\rightarrow U(L^{2}_{0}(X,\nu)).$

We also call $\pi_{0}$ a Koopman representation.

Let $\pi$ be a unitary representation of $G$ on a Hilbert space $\mathcal{H}$.
Let $\mathcal{B}(\mathcal{H})$ be the involutive algebra of bounded operators
on $\mathcal{H}$ and let $\mu$ be a finite regular Borel measure on $G$.
Recall that the operator $\pi(\mu)$ in defined in the following way:

$\pi(\mu):=\int_{G}\pi(g)\mathop{}\\!\mathrm{d}\mu(g),$

that is, $\pi(\mu)$ is the only bounded operator on $\mathcal{H}$ such that
for all $\xi,\eta\in\mathcal{H}$,

$\langle\pi(\mu)\xi,\eta\rangle:=\int_{G}\langle\pi(g)\xi,\eta\rangle\mathop{}\\!\mathrm{d}\mu(g).$

The unitary representation $\pi$ induces in this way a $*$-representation of
the involutive algebra $\mathcal{M}(G)$ of regular complex Borel measures on
$G$:

$\begin{array}[]{rcl}\mathcal{M}(G)&\rightarrow&\mathcal{B}(H)\\\
\mu&\mapsto&\displaystyle\pi(\mu).\\\ \end{array}$

See [BekDelaHa, Appendix 6]. For a regular symmetric probability measure $\mu$
on $G$, the operator $\pi(\mu)$ is self-adjoint, and its operator norm is
smaller or equal than $1$.

######  Proposition 1 (Koopman representation and discrepancies).

Consider an action of $G$ on a probability space $(X,\nu)$ by measurable
transformations preserving the measure, and let $\pi_{0}$ the sub-
representation of the Koopman representation $\pi$ on the orthogonal of the
subspace of constant functions (see (3) in Remark 1 above). Let $\mu$ be a
bounded regular Borel measure on $G$. Then $\delta(\mu)=\|\pi_{0}(\mu)\|$.

###### Proof.

Let $\phi\in L^{2}(X,\nu)$, and let us denote

$\delta(\mu,\phi):=\left\|x\mapsto\int_{G}\phi(g^{-1}x)\mathop{}\\!\mathrm{d}\mu-\int_{X}\phi\mathop{}\\!\mathrm{d}\nu\right\|_{2},$

so that

$\delta(\mu)=\sup_{\begin{subarray}{c}\phi\in L^{2}(X,\nu)\\\
\|\phi\|_{2}=1\end{subarray}}\delta(\mu,\phi).$

Let us denote $P_{1}$ the orthogonal projector on the subspace
$\mathbb{C}\textbf{1}_{X}$ of constant functions, and let $P_{0}:=I-P_{1}$.
Then it is straightforward to check that $P_{0}$ is the orthogonal projector
on $L^{2}_{0}(X)$, for every $\phi\in L^{2}(X,\nu)$,

$P_{1}(\phi)=\int_{X}\phi\mathop{}\\!\mathrm{d}\nu\cdot\textbf{1}_{X},$

that

$\delta(\mu,\phi)=\delta(\mu,P_{0}\phi)=\|\pi_{0}(\mu)P_{0}\phi\|_{2}$

and the Proposition follows. ∎

A well-known consequence of Proposition 1 is the following. Assume $\mu$ is a
Borel regular symmetric probability measure $\mu$ on $G$. If $\delta(\mu)<1$,
then there is a uniform exponential rate of convergence in the corresponding
von Neumann’s ergodic theorem; that is, we have

$\forall\phi\in L^{2}(X,\nu),\,\|\phi\|_{2}\leq
1,\,\quad\left\|\pi(\mu)^{n}\phi-\int_{X}\phi\mathop{}\\!\mathrm{d}\nu\right\|_{2}\leq\delta(\mu)^{n}.$

##  4\. Statement of the results

Let us recall that an _atom_ of a measure space $(X,\nu)$ is a measurable
subset $A$ such that $\nu(A)>0$ and every measurable subset $B$ of $A$ is such
that

$\nu(B)\in\\{0;\nu(A)\\}.$

A measure space that does not have atoms is called _atomless_.

The following proposition is a crucial technical tool in this paper.

######  Proposition 2.

Let $(X,\mathcal{B},\nu)$ be an atomless probability space. Let $G$ be a
locally compact Hausdorff group acting on $X$ by measure-preserving
transformations. We assume the corresponding Koopman representation $\pi_{0}$
is strongly continuous.

Assume for each compact symmetric neighborhood $S$ of the identity $e\in G$,
there exist a neighborhood $F$ of $e\in G$ and a sequence
$(B_{n})_{n\in\mathbb{N}}$ of measurable subsets of $X$, with the following
properties:

  1. (1)

$\forall n\in\mathbb{N},\ \nu(B_{n})>0$,

  2. (2)

$\displaystyle\inf\\{\nu(B)\mbox{ }|\mbox{ }B\in\mathcal{B},\
S^{n}B_{n}\subset B\\}<\frac{1}{2}$,

  3. (3)

$\displaystyle\limsup_{n\to\infty}\left(\inf_{g\in
F}\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\right)^{\frac{1}{n}}\geq 1$.

Then, any finite positive regular Borel measure $\mu$ on $G$ satisfies:

$\|\pi_{0}(\mu)\|\geq\|\lambda_{G}(\mu)\|.$

######  Remark 2.

The reader should notice that in condition $(2)$ above, the set $S^{n}B_{n}$
may fail to be measurable (see for example the construction, in [ERDSTONE], of
subsets $K$ and $B$ of $\mathbb{R}$ such that $K$ is compact, $B$ is a
$G_{\delta}$ \- and therefore, Borel - but $K+B$ is not Borel).

We first notice a straightforward application of the proposition (which
generalizes [PinPitExactRate, Theorem 3.6, p. 77]): if $(X,\nu)$ is an
atomless probability space, $G$ a _discrete_ group, $G\curvearrowright X$ a
measure-preserving action, and $\mu$ a probability measure on $G$, then

$\|\pi_{0}(\mu)\|\geq\|\lambda_{G}(\mu)\|.$

Indeed, in an atomless probability space, there exist measurable sets of
arbitrary small nonzero measure (see Theorem 2 below). The discretness of $G$
implies that any compact subset $S$ of $G$ is finite. Hence for any measurable
set $B_{n}$ of $X$ of small measure, namely such that
$\nu(B_{n})<\frac{1}{2|S|^{n}}$, we have
$\nu(S^{n}B_{n})\leq|S|^{n}\nu(B_{n})<\frac{1}{2}$. The discreetness of $G$
also implies that the singleton $F=\\{e\\}$ is a neighborhood of $e$. In the
case $F=\\{e\\}$, Condition (3) is obviously true for any sequence of
measurable sets of strictly positive measures.

The conclusion of Proposition 2 is sharp for all countable groups. More
precisely, it is established in [PinPitExactRate, Corollary 3.12], that for
every countable group $G$, there is a measurable action (namely, the Bernoulli
shift on $G$) such that for each finitely-supported probability measure $\mu$
on $G$, $\|\lambda_{G}(\mu)\|=\delta(\mu)$.

We do not know if every locally compact Hausdorff group admits an action with
a measure with discrepancies equal to the lower bounds of Proposition 2.

In Proposition 3 below, we invoke Definition 6.1 and prove that Proposition 2
implies the following theorem.

######  Theorem 1.

Let $X$ be a topological space, $\nu$ be an atomless Borel probability measure
on $X$, and $G\curvearrowright X$ a continuous action of a locally compact
Hausdorff unimodular group that preserves $\nu$. We assume the corresponding
Koopman representation $\pi_{0}$ is strongly continuous.

Suppose there exists a point $x_{0}$ in the support of $\nu$ such that for any
compact subset $K$ of $G$ we have $\nu(Kx_{0})=0$ and
$G_{x_{0}}=\operatorname{Stab}_{G}(x_{0})\mbox{ is compact}$. Then, any finite
positive regular Borel measure $\mu$ on $G$ satisfies:

$\|\pi_{0}(\mu)\|\geq\|\lambda_{G}(\mu)\|.$

The following corollary is an application of the theorem (see Subsection 6.3
below for details).

######  Corollary 1.

Let $G$ be a locally compact Hausdorff group, $\mu_{G}$ be a left Haar measure
on $G$, and $H$ a unimodular closed subgroup of $G$ such that $\mu_{G}(H)=0$.
Let $\Gamma$ be a lattice in $G$, such that $H\cap\Gamma=\\{e\\}$. Let
$\mu_{G/\Gamma}$ be the unique $G$-invariant Borel regular probability measure
on $G/\Gamma$. We consider the action $H\curvearrowright G/\Gamma$, that
preserves $\mu_{G/\Gamma}$. Then, any finite positive regular Borel measure
$\mu$ on $H$ satisfies:

$\|\pi_{0}(\mu)\|\geq\|\lambda_{H}(\mu)\|.$

Recall that if $G$ is amenable and $\mu$ is a regular symmetric probability
measure, whose support generates $G$, then $\|\lambda_{G}(\mu)\|=1,$ (see
[Kes] for the case $G$ is discrete and [PIER] for the general case). Hence, if
$G$ is discrete and amenable, $(X,\nu)$ is an atomless probability space and
$G\curvearrowright X$ is a measurable action which preserves $\nu$, then the
discrepancy of any symmetric probability measure on $G$ equals $1$ (see
[PinPitExactRate, Corollary 3.10]). In the same vein, since $\mathbb{R}$ is
unimodular and amenable, Theorem 1 implies that for every continuous
$\mathbb{R}$-action by measure-preserving transformations with negligible
orbits, the discrepancy of any compactly-supported, nonnegative, continuous
function is at least $1$. For example, this yields a proof of the following
folklore statement (although the result is known to experts in the field, we
have not been able to localize a published proof): let $S$ be a compact,
hyperbolic surface. Consider $g_{t}:T_{1}S\rightarrow T_{1}S$ the geodesic
flow on the unit tangent bundle, and let $\nu$ be the Liouville measure on
$T_{1}S$. Then, for every $T\in\mathbb{R}^{*}$,

$\sup_{\begin{subarray}{c}\phi\in L^{2}(T^{1}S,\nu)\\\
\|\phi\|_{2}=1\end{subarray}}\left\|\left(x\longrightarrow\frac{1}{T}\int^{T}_{0}\phi(g_{t}x)\mathop{}\\!\mathrm{d}t-\int_{T_{1}S}\phi\mathop{}\\!\mathrm{d}\nu\right)\right\|_{2}=1.$

The same proof applies to the geodesic flow of a non-compact locally symmetric
space.

There are non-amenable examples where the inequality in Corollary 1 is an
equality, and we describe one of them in Section 7.2.

##  5\. Interpretation of the technical condition in Proposition 2

We give a heuristic example to illustrate the technical condition, i.e.
Condition (3), in Proposition 2 that we postulate in order to be able to treat
actions of locally compact non-discrete groups.

######  Example 1.

Let us consider the action of $\mathbb{R}$ on
$\mathbb{T}^{2}:=(\mathbb{R}/\mathbb{Z})^{2}$ by translations along a line $d$
with irrational slope; that is, let $\alpha\in\mathbb{R}\setminus\mathbb{Q}$,
and define, for all $t\in\mathbb{R}$, $(a,b)\in\mathbb{T}^{2}$,

$(a+t,b+\alpha t).$

Let $F:=[-1,1]$. On Figure 1, we represent $\mathbb{R}^{2}$ twice, and on each
of the two pictures, we identify $\mathbb{T}^{2}$ with $[0,1]^{2}$. Let us
consider two isometric long rectangles, centered at the origin, in
$\mathbb{R}^{2}$, such that two opposite sides are parallel to $d$. Notice
that the two opposite sides parallel to $d$ are short in the case of the red
rectangle and are long in the case of the blue rectangle. Consider conditions
$(1),(2),(3),$ from Proposition 2. Let $g$ be a small translation (i.e. with
amplitude in $F$) along the line $d$ (drawn in black with a small arrow
suggesting the action of $g$). Let $R_{red}$ be the projection of the red
rectangle, and $R_{blue}$ be the projection of the blue rectangle. These two
projected rectangles represent possible choices of $B_{n}^{\prime}s$ in the
statement of Proposition 2. One can graphically expect that $gR_{red}\cap
R_{red}$ is quite small, whereas $gR_{blue}\cap R_{blue}$ is approximately as
big as $R_{blue}$. One can therefore think of sequences $(B_{n})_{n}$ that
satisfy Condition (3) in Proposition 2 as subsets that look like $R_{blue}$.

Figure 1. The opaque-coloured rectangles are the projections of the
transparent ones. The segment in black is a subset of $d$.

##  6\. Proofs

We first recall a theorem of Sierpiński on atomless probability spaces we will
need in the proof of Proposition 2.

######  Theorem 2 (See [Sierpinski]).

Let $(X,\mathcal{B},\nu)$ be an atomless probability space. Then $(X,\nu)$ has
the _intermediate value property_ , that is: for all measurable subsets $A,B$
of $X$ such that $A\subset B$, for all $c\in[\nu(A),\nu(B)]$, there exists a
measurable subset $C$ of $X$ such that

  * •

$A\subset C\subset B$;

  * •

$\nu(C)=c$.

(A stronger statement can be proved using Zorn’s lemma: for all such $A,B$,
there exists a map $f:[\nu(A),\nu(B)]\rightarrow\mathcal{B}$ such that $f$ is
$(\leq,\subset)$-order-preserving and for all $c\in[\nu(A),\nu(B)]$,
$\nu(f(c))=c$.)

###  6.1. Proof of Proposition 2

We first state the following definition.

######  Definition 6.1 (Moderate growth).

Let $G$ be any group, $(X,\mathcal{B},\nu)$ an atomless probability space, and
$G\curvearrowright X$ a measure-preserving action. Let $F,S\subset G$ be
measurable subsets. We say that a sequence $(B_{n})_{n\in\mathbb{N}}$ of
measurable subsets of $X$ is of $(S,F)$-moderate growth if

  1. (1)

$\forall n\in\mathbb{N},\ \nu(B_{n})>0$,

  2. (2)

$\displaystyle\inf\\{\nu(B)\mbox{ }|\mbox{ }B\in\mathcal{B},\
S^{n}B_{n}\subset B\\}<\frac{1}{2}$,

  3. (3)

$\displaystyle\limsup_{n\to\infty}\left(\inf_{g\in
F}\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\right)^{\frac{1}{n}}\geq 1$.

Proposition 2 therefore states that if for any compact symmetric neighborhood
$S$ of $e$ in $G$, there exists a neighborhood $F$ of $e$ in $G$ and an
$(S,F)$-moderate growth sequence, then Inequality (1) holds.

###### Proof of Proposition 2.

We prove the proposition in several steps.

Step 1: we assume _$\mu$ is of positive type, with support $S$ a compact
symmetric neighborhood of the identity $e\in G$_. As $G$ is locally compact,
we may assume the neighborhood $F$ of $e$ given by the hypothesis of the
proposition is compact. Let $(B^{+}_{n})_{n\in\mathbb{N}}$ be a sequence of
$F$-moderate growth. Atomless probability spaces satisfy the so-called
_intermediate value property_ (see Theorem 2 above). Hence, there exists a
subset $B^{-}_{n}$ that has the same $\nu$-measure as $B_{n}$, and such that

(2) $S^{n}B_{n}\cap B^{-}_{n}=\emptyset,$

and as $S$ is symmetric the same equality also holds with $B_{n}$ and
$B^{-}_{n}$ exchanged. Let

$\phi:=\frac{\mathbf{1}_{B_{n}}-\mathbf{1}_{B^{-}_{n}}}{\|\mathbf{1}_{B_{n}}-\mathbf{1}_{B^{-}_{n}}\|_{2}}.$

The function $\phi$ is obviously in $L^{2}_{0}(X,\nu)$. Let $\mu^{*n}$ be the
$n$-fold convolution of $\mu$ with itself. Notice that

$S^{n}=(\mbox{support}(\mu))^{n}=\mbox{support}(\mu^{*n}).$

We therefore have,

$\begin{array}[]{rcl}\|\pi_{0}(\mu^{*n})\|&\geq&\langle\pi_{0}(\mu^{*n})\phi,\phi\rangle\\\
&=&\displaystyle\frac{1}{\nu(B_{n})+\nu(B^{-}_{n})}\int_{G}\left[\nu\left(gB_{n}\cap
B_{n}\right)+\nu\left(gB^{-}_{n}\cap
B^{-}_{n}\right)\right]\mathop{}\\!\mathrm{d}\mu^{*n}(g)\\\
&&-\displaystyle\frac{1}{\nu(B_{n})+\nu(B^{-}_{n})}\int_{G}\left[\nu\left(gB_{n}\cap
B^{-}_{n}\right)+\nu\left(gB^{-}_{n}\cap
B_{n}\right)\right]\mathop{}\\!\mathrm{d}\mu^{*n}(g)\\\
&\stackrel{{\scriptstyle\eqref{b+b-
disjoint}}}{{=}}&\displaystyle\frac{1}{\nu(B_{n})+\nu(B^{-}_{n})}\int_{G}\left[\nu\left(gB_{n}\cap
B_{n}\right)+\nu\left(gB^{-}_{n}\cap
B^{-}_{n}\right)\right]\mathop{}\\!\mathrm{d}\mu^{*n}(g)\\\
&\geq&\displaystyle\frac{1}{\nu(B_{n})+\nu(B^{-}_{n})}\int_{F}\left[\nu\left(gB_{n}\cap
B_{n}\right)+\nu\left(gB^{-}_{n}\cap
B^{-}_{n}\right)\right]\mathop{}\\!\mathrm{d}\mu^{*n}(g)\\\
&\geq&\displaystyle\frac{1}{\nu(B_{n})+\nu(B^{-}_{n})}\inf_{g\in
F}\nu(gB_{n}\cap B_{n})\int_{F}\mathop{}\\!\mathrm{d}\mu^{*n}(g)\\\
&=&\displaystyle\frac{1}{2}\mu^{*n}(F)\inf_{g\in
F}\displaystyle\frac{\nu(gB_{n}\cap B_{n})}{\nu(B_{n})}.\\\ \end{array}$

We know that

$\|\pi_{0}(\mu)\|=\limsup_{n\to\infty}\|\pi_{0}(\mu)^{n}\|^{\frac{1}{n}}.$

As $\mu$ is symmetric and $F$ is a compact neighborhood of the identity, it
follows from [Berg-Christensen], that

$\limsup_{n\to\infty}(\mu^{*n}(F))^{\frac{1}{n}}=\|\lambda_{G}(\mu)\|.$

Furthermore, the hypothesis that $\mu$ is of positive type implies that the
above lim sup is in fact a genuine limit [Berg-Christensen]. We can now finish
the proof of Step 1:

$\begin{array}[]{rcl}\|\pi_{0}(\mu)\|&=&\limsup_{n\to\infty}\|\pi_{0}(\mu)^{n}\|^{\frac{1}{n}}\\\
&\geq&\limsup_{n\to\infty}\frac{1}{2}^{\frac{1}{n}}\mu^{*n}(F)^{\frac{1}{n}}\left(\inf_{g\in
F}\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\right)^{\frac{1}{n}}\\\
&=&\|\lambda_{G}(\mu)\|\limsup_{n\to\infty}\left(\inf_{g\in
F}\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\right)^{\frac{1}{n}}\\\
&\geq&\|\lambda_{G}(\mu)\|.\end{array}$

Step 2: we assume _the support of $\mu$ is a compact neighborhood of the
identity_. The measure $\eta=\mu*\mu^{*}$ satisfies the hypothesis of Step 1.
For any unitary representation $\alpha$ of $G$,
$\|\alpha(\eta)\|=\|\alpha(\mu)\|^{2}$. Hence Step 2 follows from Step 1.

Step 3: we assume _the support of $\mu$ is a neighborhood of the identity_. As
$G$ is locally compact we may choose $K_{1}$, a compact neighborhood of the
identity, included in the support of $\mu$. As $\mu$ is regular and finite, we
may choose a increasing sequence $K_{n}\subset K_{n+1}$ of compact subsets of
$G$ such that $\mu(K_{n})$ converges to $\mu(G)$ as $n\to\infty$, and
approximate $\mu$ with the truncations $\mu_{n}={\mathbb{1}}_{K_{n}}\cdot\mu$.
As $\mu$ and $\mu_{n}$ are finite regular we may apply [Godement, 6.
Opérateurs de Radon] to get, for any unitary representation $\alpha$ of $G$,

$|\|\alpha(\mu)\|-\|\alpha(\mu_{n})\||\leq\mu(G)-\mu(K_{n}).$

Step 2 applies to $\mu_{n}$ for any $n$. Hence Step 3 follows from Step 2.

Step 4: we assume _the support of the measure $\mu$ has positive Haar
measure_. We consider $\eta=\mu*\mu^{*}$. As the Haar measure of the support
of $\mu$ is strictly positive,

$\mbox{support}(\eta)=\overline{(\mbox{support}(\mu))\cdot(\mbox{support}(\mu))^{-1}}$

is a neighborhood of the identity according to [Weil, page 50]. (In order to
prove the inclusion
$\mbox{support}(\mu*\mu^{*})\subset\overline{\mbox{support}(\mu)\cdot\mbox{support}(\mu^{*})}$,
we use that $\mu$ is inner regular. The other inclusion is true for any Borel
measure on any topological group.) Again, for any unitary representation
$\alpha$ of $G$, $\|\alpha(\eta)\|=\|\alpha(\mu)\|^{2}$. Hence Step 4 follows
from Step 3.

Step 5: we assume the measure $\mu$ is as in the proposition. We choose a
compact neighborhood $K$ of $e\in G$ and define, for any integer
$n\in\mathbb{N}$, the measure
$\mu_{n}=\mu+\frac{1}{n}{\mathbb{1}}_{K}\cdot\mu_{G}$. By construction, the
support of $\mu_{n}$ has positive Haar measure. Moreover,
$\lim_{n\to\infty}|\mu_{n}(G)-\mu(G)|=0$. Hence Step 5 follows from Step 4.

∎

###  6.2. Nets and volume estimates

The goal of this subsection is to establish counting inequalities that allow
us to estimate the cardinal of maximal nets, that is, finite sets of evenly-
spaced points, in big subsets (that can be thought as balls). The precise
technical statement that will be used later in the proofs is Lemma 5.

######  Lemma 1.

Let $G$ be a topological group, $A$ and $B$ two compact subsets, such that $B$
is a neighborhood of $e$. Then, for all $k\in\mathbb{N}$, there exists a
finite subset $S\subset G$ such that we have, for all $n\in\mathbb{N}^{*}$,

$B^{k+n-1}A\subset B^{n}S.$

###### Proof.

Let $k\in\mathbb{N}$. Let $U$ be an open set in $G$ containing $e$ and
contained in $B$. Then the compact subset $B^{k}A$, is covered by the sets
$Uh$ for $h\in B^{k}A$. Therefore, there is a finite subset $S$ of $B^{k}A$
such that $B^{k}A\subset US$. Since $US\subset BS$, we therefore have
$B^{k}A\subset BS$, hence, for all $n\in\mathbb{N}^{*}$, $B^{k+n-1}A\subset
B^{n}S$. ∎

Note that in the following Lemma, we consider a right-Haar measure.

######  Lemma 2.

Let $G$ be a locally compact Hausdorff group, let $\mu_{G}$ be a right-Haar
measure on $G$, let $A,B$ be compact subsets of $G$, such that $B$ is a
neighborhood of $e$. Then, for all $k\in\mathbb{N}$, there is a constant $c$
such that

$\forall n\in\mathbb{N}^{*},\quad\mu_{G}(B^{n+k-1}A)\leq c\mu_{G}(B^{n}).$

###### Proof.

The claim follows immediately from Lemma 1 : it is enough to take $c:=|S|$
where $S$ is a finite subset given by Lemma 1. ∎

######  Definition 6.2 (Net).

Let $G$ be any group. Let $A,B,N\subset G$. We say that $N$ is an
$A$-separated net of $B$ if

  1. (1)

$N$ is a finite subset of $B$ ;

  2. (2)

$\forall n_{1},n_{2}\in N,\quad n_{1}A\cap n_{2}A\neq\emptyset\Rightarrow
n_{1}=n_{2}$.

Let us denote by $N(B,A)$ the set of $A$-separated nets in $B$. We endow
$N(B,A)$ with the order given by inclusion and denote by $N_{max}(B,A)$ the
set of maximal nets (that is, nets that are not strictly contained in a bigger
net).

######  Lemma 3.

Let $G$ be any group, and let $A,B\subset G$. We have

$\forall N\in N_{max}(B,A),\quad B\subset NAA^{-1}.$

###### Proof.

Let $N\in N_{max}(A,B)$. Let us show that we have $B\subset NAA^{-1}$. We
proceed by contradiction. Assume that $b\in B\setminus NAA^{-1}$. Let us show
that $N\cup\\{b\\}$ is an $A$-separated net of $B$. It is enough to check that
for every $n\in N$, $nA\cap bA=\emptyset$. Let $n\in N$. If $nA\cap
bA\neq\emptyset$, there exists $a_{1},a_{2}\in A$ such that $na_{1}=ba_{2}$.
Let $a_{1},a_{2}$ as such. Then $b=na_{1}a^{-1}_{2}\in NAA^{-1}$, which is a
contradiction. So $nA\cap bA=\emptyset$. Therefore, $N\cup\\{b\\}$ is an
$A$-separated net of $B$, and this contradicts the maximality of $N$. So we
have $B\subset NAA^{-1}$. ∎

######  Lemma 4.

Let $G$ be a locally compact Hausdorff group, and $\mu_{G}$ be a left-Haar
measure on $G$. Let $A,B\subset G$ such that $A$ $B$ and $BA$ are measurable.
We then have

$\forall N\in N(B,A),\quad|N|\mu_{G}(A)\leq\mu_{G}(BA),$

and

$\forall N\in N_{max}(B,A),\quad\mu_{G}(B)\leq|N|\mu_{G}(AA^{-1}).$

###### Proof.

Let us prove the first item. From the definition of a net, $NA$ is the
disjoint union of the $nA$ for $n$ in $N$, so $\mu_{G}(NA)=|N|\mu_{G}(A)$. But
since $N\subset B$, $NA\subset BA$, so we get the desired inequality. Let us
now prove the second item. From Lemma 3, $B\subset NAA^{-1}$. So, we have
$\mu_{G}(B)\leq\mu_{G}(NAA^{-1})\leq\mu_{G}(AA^{-1})|N|$. ∎

######  Lemma 5.

Let $G$ be a locally compact Haudorff unimodular group, $A_{1},A_{2},B,$
compact neighborhoods of $e$ in $G$. Then there are constants $c_{1},c_{2}>0$
such that

$\forall n\geq 3,\ \forall N_{1}\in N_{max}(B^{n},A_{1}),\forall N_{2}\in
N_{max}(B^{n-2},A_{2}),\quad c_{1}\leq\displaystyle\frac{|N_{2}|}{|N_{1}|}\leq
c_{2}.$

###### Proof.

From Lemma 2, there are strictly positive $k_{1},k_{2},k_{3}$ such that

(3) $\forall n\in\mathbb{N},\quad\mu_{G}(B^{n}A_{1})\leq k_{1}\mu_{G}(B^{n}),$
(4) $\forall n\geq 3,\quad\mu_{G}(B^{n-2}A_{2})\leq k_{2}\mu_{G}(B^{n}),$ (5)
$\forall n\in\mathbb{N},\quad\mu_{G}(B^{n+2})\leq k_{3}\mu_{G}(B^{n}).$

Let $n\geq 3$, $N_{1}\in N_{max}(B^{n},A_{1})$, $N_{2}\in
N_{max}(B^{n-2},A_{2})$. We have

(6) $|N_{2}|\stackrel{{\scriptstyle Lemma\ \ref{cardinal filet
mesure}}}{{\leq}}\frac{\mu_{G}(B^{n-2}A_{2})}{\mu_{G}(A_{2})}\stackrel{{\scriptstyle\eqref{eq:
k_2}}}{{\leq}}\displaystyle\frac{k_{2}}{\mu_{G}(A_{2})}\mu_{G}(B^{n})$

Moreover,

(7) $\mu_{G}(B^{n-2})\frac{1}{\mu_{G}(A_{2}A^{-1}_{2})}\stackrel{{\scriptstyle
Lemma\ \ref{cardinal filet mesure}}}{{\leq}}|N_{2}|.$

So, we have

$\displaystyle\frac{1}{k_{3}\mu_{G}(A_{2}A^{-1}_{2})}\mu_{G}(B^{n})\stackrel{{\scriptstyle\eqref{eq:
k_3}}}{{\leq}}\mu_{G}(B^{n-2})\displaystyle\frac{1}{\mu_{G}(A_{2}A^{-1}_{2})}\stackrel{{\scriptstyle\eqref{eq:
minoration N-2n}}}{{\leq}}|N_{2}|\stackrel{{\scriptstyle\eqref{eq: majoration
N-2n}}}{{\leq}}\displaystyle\frac{k_{2}}{\mu_{G}(A_{2})}\mu_{G}(B^{n}).$

On the other hand,

$|N_{1}|\stackrel{{\scriptstyle Lemma\ \ref{cardinal filet
mesure}}}{{\leq}}\frac{\mu_{G}(B^{n}A_{1})}{\mu_{G}(A_{1})},$

so

$|N_{1}|\stackrel{{\scriptstyle\eqref{eq:
k_1}}}{{\leq}}\frac{k_{1}}{\mu_{G}(A_{1})}\mu_{G}(B^{n}).$

Moreover,

$\mu_{G}(B^{n})\frac{1}{\mu_{G}(A_{1}A^{-1}_{1})}\stackrel{{\scriptstyle
Lemma\ \ref{lem estimation cardinal des filets}}}{{\leq}}|N_{1}|.$

So we have

$\mu_{G}(B^{n})\frac{1}{\mu_{G}(A_{1}A^{-1}_{1})}\leq|N_{1}|\leq\frac{k_{1}}{\mu_{G}(A_{1})}\mu_{G}(B^{n}).$

We therefore obtain

$\frac{\mu_{G}(A_{1})}{k_{3}k_{1}\mu_{G}(A_{2}A^{-1}_{2})}\leq\frac{|N_{2}|}{|N_{1}|}\leq\frac{k_{2}\mu_{G}(A_{1}A^{-1}_{1})}{\mu_{G}(A_{2})}.$

∎

###  6.3. Proof of Theorem 1 and Corollary 1

In order to prove Theorem 1, it is obviously enough to prove the following
proposition:

######  Proposition 3.

Let $G$ be a locally compact Hausdorff unimodular group. Let $X$ be a
Hausdorff topological space, $\nu$ an atomless regular Borel measure on $X$.
Consider a continuous measure-preserving action $G\curvearrowright X$. We
assume that there exists $x_{0}$ in the support of $\nu$ such that
$\operatorname{Stab}_{G}(x_{0})$ is compact and for any compact subset $K$ of
$G$, $\nu(Kx_{0})=0$. Then for each compact symmetric neighborhood $S$ of $e$
in $G$, there exists a compact, symmetric neighborhood $F$ of $e$, and a
sequence of neighborhoods of $x_{0}$ that is of $(S,F)$-moderate growth.

######  Remark 3.

If $G$ is $\sigma$-compact, then $\nu(Kx)=0$ for every compact $K$ if and only
if the orbit of $x_{0}$ is a null-set.

We will need the following lemma.

######  Lemma 6.

Let $G$ be a topological group, $X$ a Hausdorff topological space. Let
$G\curvearrowright X$ be a continuous action (that is, the map $G\times
X\rightarrow X$ that defines it is continuous).

Let $x_{0}\in X$, let $K$ be a compact subset of $G$, and $U$ a neighborhood
of $e$ in $G$. Then there exists a neighborhood $V$ of $x_{0}$ in $X$ such
that

$\forall g\in K\setminus\operatorname{Stab}_{G}(x_{0})U,\quad gV\cap
V=\emptyset.$

###### Proof.

First of all, let us notice that $\operatorname{Stab}_{G}(x_{0})U$ is open,
and therefore $K\setminus\operatorname{Stab}_{G}(x_{0})U$ is compact.

Let $E$ be the set of triples $(g,W,V)$ such that

  * •

$g\in K\setminus\operatorname{Stab}_{G}(x_{0})U$;

  * •

$W$ is a neighborhood of $g$ in $G$;

  * •

$V$ is a neighborhood of $x_{0}$ in $X$;

  * •

$\forall w\in W,\quad wV\cap V=\emptyset$.

For all $g\in K\setminus\operatorname{Stab}_{G}(x_{0})U$, there exists $W,V$
such that $(g,W,V)\in E$. Indeed, $g\not\in\operatorname{Stab}_{G}(x_{0})$, so
$gx_{0}\neq x_{0}$. Let $Y$ be a closed neighborhood of $x_{0}$ that does not
contain $gx_{0}$. By continuity, the set of $(h,x)$ such that $hx\not\in Y$ is
an open subset of $G\times X$; moreover, it contains $(g,x_{0})$, so there are
neighborhoods $W,Z$ of $g$ and $x_{0}$ such that $WZ\subset Y^{c}$. Let
$V:=Z\cap Y$. Then $WV\subset Y^{c}\subset V^{c}$, so $WV\cap V=\emptyset$.

Therefore, the set of second coordinates of $E$ covers
$K\setminus\operatorname{Stab}_{G}(x_{0})U$. By compacity, there exists
$n\in\mathbb{N}^{*}$, $g_{1},\cdots,g_{n}$, $W_{1},\cdots,W_{n}$,
$V_{1},\cdots,V_{n}$ such that
$K\setminus\operatorname{Stab}_{G}(x_{0})U\subset\bigcup_{i}W_{i}$. Let us
define $V:=\bigcap_{i}V_{i}$. This subset $V$ satisfies the needed
requirements: it is a neighborhood of $x_{0}$, and if $g\in
K\setminus\operatorname{Stab}_{G}(x_{0})U$, then there exists $i$ such that
$g\in W_{i}$. Therefore $gV_{i}\cap V_{i}=\emptyset$, so $gV\cap V=\emptyset$.
∎

###### Proof of Proposition 3.

Let $S$ be a compact, symmetric neighborhood of $G$. We would like to find a
compact symmetric neighborhood $F$ of $e$ and a sequence $(B_{n})_{n\geq 3}$
that has $(S,F)$-moderate growth. Let $n\geq 3$.

First of all, since $\nu(S^{2n}x_{0})=0$, and since $\nu$ is regular, there
exists an open subset $U$ of $X$ containing $S^{2n}x_{0}$, that has measure
strictly smaller than $\frac{1}{2}$.

We claim that there exists a neighborhood $W_{n}$ of $x_{0}$ such that
$S^{2n}W_{n}\subset U$. Indeed, by continuity of the action, for all $g\in
S^{2n}$, there exists a neighborhood $D_{g}$ of $g$ and a neighborhood $E_{g}$
of $x_{0}$ such that $D_{g}E_{g}\subset U$. By compactness of $S^{2n}$, there
exists a finite subset $I$ of $S^{2n}$ such that $S^{2n}\subset\bigcup_{i\in
I}D_{i}$. Let us now define $W_{n}:=\bigcap_{i\in I}E_{i}$. We have

$S^{2n}W_{n}\subset\bigcup_{i\in I}D_{i}W_{n}\subset\bigcup_{i\in
I}D_{i}E_{i}\subset U,$

which gives the claim.

Now, taking $F=S$, according to Lemma 6, there exists a neighborhood $V_{n}$
of $x_{0}$ such that

(8) $\forall g\in S^{2n-2}\setminus\operatorname{Stab}_{G}(x_{0})S,\quad
gV_{n}\cap V_{n}=\emptyset;$

moreover, up to taking a smaller neighborhood instead of $V_{n}$, we can
assume that $V_{n}$ is contained in $W_{n}$.

Let us then define

$B_{n}:=S^{n}V_{n}.$

We now check that the sequence $(B_{n})_{n\geq}$ fulfills the three
requirements of $(S,F)$-moderate growth.

First of all, since $V_{n}$ is a neighborhood of $x_{0}$ and since $x_{0}$ is
in the support of $\nu$, $\nu(B_{n})>0$, so the first condition is verified.
Moreover, by construction, $S^{n}B_{n}=S^{2n}V_{n}\subset U$; and
$\nu(U)<\frac{1}{2}$ so the second condition is satisfied.

We will now estimate, for all $g\in F$,

$\displaystyle\frac{\nu(gB_{n}\cap B_{n})}{\nu(B_{n})}$

by considering nets.

Let $N_{1}\in N_{max}(S^{n-2},S\operatorname{Stab}_{G}(x_{0})S^{2})$. We have
$N_{1}F\subset F^{n-1}$, so

(9) $\nu(N_{1}FV_{n})\leq\nu(F^{n-1}V_{n}).$

Moreover, we have

$N_{1}FV_{n}=\bigsqcup_{g\in N_{1}}gFV_{n};$

indeed, on the one hand, let $g_{1},g_{2}\in N_{1},\ f_{1},f_{2}\in F,\
v_{1},v_{2}\in V_{n}$ such that $g_{1}f_{1}v_{1}=g_{2}f_{2}v_{2}$. We then
have $f^{-1}_{2}g^{-1}_{2}g_{1}f_{1}v_{1}=v_{2}$. On the other hand,
$f^{-1}_{2}g^{-1}_{2}g_{1}f_{1}\in F^{2n-2}$, so, by condition (8) on $V_{n}$,
we have $f^{-1}_{2}g^{-1}_{2}g_{1}f_{1}\in\operatorname{Stab}_{G}(x_{0})F$.
So, $g_{1}\in g_{2}F\operatorname{Stab}_{G}(x_{0})F^{2}$, and this is only
possible if $g_{1}=g_{2}$.

We deduce from this

$\nu(N_{1}FV_{n})=|N_{1}|\nu(FV_{n}),$

so

$|N_{1}|\nu(FV_{n})\stackrel{{\scriptstyle\eqref{eq:
xxx}}}{{\leq}}\nu(F^{n-1}V_{n}).$

Now, for any $g\in F$, we have $F^{n-1}V_{n}\subset gF^{n}V_{n}\cap
F^{n}V_{n}=gB_{n}\cap B_{n}$, so

$\nu(gB_{n}\cap B_{n})\geq|N_{1}|\nu(FV_{n}).$

Finally, let $C$ be a compact neighborhood of $e$ in $G$ such that
$CC^{-1}\subset F$. Let $N_{2}\in N_{max}(F^{n},C)$. Then

$B_{n}=F^{n}V_{n}\stackrel{{\scriptstyle Lemma\ \ref{lem reunion
filet}}}{{\subset}}N_{2}CC^{-1}V_{n}$

so

$\nu(B_{n})\leq|N_{2}|\nu(FV_{n}).$

We therefore have, for all $n\geq 3$,

$\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\geq\displaystyle\frac{|N_{1}|}{|N_{2}|},$

so

$\left(\displaystyle\frac{\nu(gB_{n}\cap
B_{n})}{\nu(B_{n})}\right)^{\frac{1}{n}}\geq\left(\displaystyle\frac{|N_{1}|}{|N_{2}|}\right)^{\frac{1}{n}}.$

The third requirement follows now from Lemma 5. ∎

We now add a few words to show that Corollary 1 follows in a straightforward
way from Theorem 1.

###### Proof of Corollary 1.

It is immediate to check the hypotheses of Theorem 1 are satisfied: let $G$ be
a locally compact group, $H$ a closed subgroup of $G$ such that
$\mu_{G}(H)=0$, and $\Gamma$ a lattice in $G$ such that $H\cap\Gamma=\\{e\\}$.
The action of $H$ on $G/\Gamma$ by left-translations is continuous, its orbits
are null-sets, and the stabilizer $H\cap\Gamma$ of the coset $e\Gamma$ is
finite, hence compact. ∎

##  7\. Examples and counterexamples

###  7.1. Margulis’ counterexample

This section, provides many examples showing that Theorem [PinPitExactRate,
Thm 3.6] cannot extend to the case where $G$ is locally compact without making
further hypotheses.

######  Example 2.

Consider the action by left-translation of $\mathbb{R}$ on
$\mathbb{R}/\mathbb{Z}$. Let, for $t\in\mathbb{R}$, $\pi_{0}(t)$ be the
operator that sends every zero-integral $\phi\in
L^{2}(\mathbb{R}/\mathbb{Z},Leb)$ to the map $s\mapsto\phi(s-t)$. Let $\mu$ be
the uniform probability measure on $[0,1]$. Then, obviously, $\pi_{0}(\mu)$ is
the zero operator, whereas $\lambda_{\mathbb{R}}(\mu)$ is not.

In fact, we have the following generalization, that follows from the
discussion that one can find in pages $107-112$ of the book [MARGU]. We give a
sketch of the proof for the convenience of the reader.

######  Proposition 4 (Margulis).

Let $G$ be a locally compact Hausdorff $\sigma$-compact group, and let $H$ be
a closed cocompact subgroup of $G$. We consider the action by left-
multiplication of $G$ on $G/H$ and we suppose that there is a $G$-invariant
probability measure $\nu$ on $G/H$. Let $\pi$ denote the associated Koopman
representation, and $\pi_{0}$ the sub-representation on the subspace of zero
integral. Then there is a continuous, compactly-supported, nonnegative
function $f$ on $G$ that has integral $1$, such that

$\|\pi_{0}(f)\|<1.$

###### Proof sketch.

Assume, by contradiction, that for every continuous, compactly-supported,
nonnegative function $f$ on $G$ that has integral $1$, we have

$\|\pi_{0}(f)\|=1.$

From [MARGU, (1.3) Proposition, p. 109], there exists a sequence
$(p_{i})_{i\in\mathbb{N}}$ that is asymptotically $\pi_{0}$-invariant, that
is, each $p_{i}$ is an element of $L^{2}_{0}(G/H)$, has norm $1$, and we have,
for every compact $K$ in $G$,

$\lim_{i\to\infty}\sup_{g\in K}\|\pi_{0}(g)p_{i}-p_{i}\|_{2}=0.$

Let $(p_{i})_{i\in\mathbb{N}}$ be such a sequence. First of all, we have

$\begin{array}[]{rcl}\displaystyle\limsup_{i\to\infty}\|\pi_{0}(f)p_{i}-p_{i}\|_{2}&\leq&\displaystyle\limsup_{i\to\infty}\sup_{g\in\operatorname{supp}(f)}\|\pi_{0}(g)p_{i}-p_{i}\|_{2}\\\
&=&0\\\ \end{array}$

Let $K$ be a compact subset of $G$. We then have

$\begin{array}[]{rcl}\displaystyle\limsup_{i\to\infty}\ \sup_{g\in
K}\|\pi_{0}(g)\pi_{0}(f)p_{i}-\pi_{0}(f)p_{i}\|_{2}&\leq&\displaystyle\limsup_{i\to\infty}\
\sup_{g\in K}\left(\|\pi_{0}(g)\pi_{0}(f)p_{i}-\pi_{0}(g)p_{i}\|_{2}\right.\\\
&&\quad\displaystyle+\left.\|\pi_{0}(g)p_{i}-p_{i}\|_{2}+\|p_{i}-\pi_{0}(f)p_{i}\|_{2}\right)\\\
&=&\displaystyle\limsup_{i\to\infty}\ \sup_{g\in
K}\left(2\|\pi_{0}(f)p_{i}-p_{i}\|_{2}\right.\\\
&&\quad+\left.\|\pi_{0}(g)p_{i}-p_{i}\|_{2}\right)\\\ &=&0\\\ \end{array}$

so the sequence

$\left(\displaystyle\frac{\pi_{0}(f)p_{i}}{\|\pi_{0}(f)p_{i}\|_{2}}\right)_{i\in\mathbb{N}}$

is also asymptotically $\pi_{0}$-invariant. Let us denote, for all
$i\in\mathbb{N}$, $q_{i}$ the composition of

$\displaystyle\frac{\pi_{0}(f)p_{i}}{\|\pi_{0}(f)p_{i}\|_{2}}$

with the canonical surjection $G\rightarrow G/H$. So, from [MARGU, (1.7)
Lemma, p. 110], the sequence $(q_{i})_{i\in\mathbb{N}}$ is equicontinuous and
uniformly bounded on $G$. From Ascoli’s theorem, up to extraction, one can
assume that it converges uniformly on compact subsets to a continuous,
$H$-invariant on the right function (since each term of the sequence is
$H$-invariant on the right). This limit gives, on the quotient $G/H$, a
function that we denote by $p$. Since $G/H$ is compact, we have the uniform
convergence

$\displaystyle\frac{\pi_{0}(f)p_{i}}{\|\pi_{0}(f)p_{i}\|_{2}}\rightarrow p.$

We deduce from all this

  * •

that $p$ has zero integral (since it is the limit of a sequence of zero-
integral functions);

  * •

that $p$ has $L^{2}$-norm $1$ (as a uniform limit of uniformly bounded
functions of $L^{2}$-norm $1$);

  * •

that $p$ is $\pi_{0}$-invariant (as a uniform limit of $\pi_{0}$-invariant
functions) and therefore is constant.

So, $p$ is identically $0$, but has $L^{2}$-norm $1$. This is a contradiction.
∎

######  Remark 4.

In particular, if $G$ is a locally compact, $\sigma$-compact amenable group,
if $H$ is a cocompact closed subgroup of $G$ such that $G\curvearrowright G/H$
has an invariant probability measure, and if $\pi_{0}$ denotes the sub-
representation of the Koopman representation on the subspace of zero-integral
functions, then there is a continuous, positive, symmetric function $h$ of
integral $1$ such that

$\|\pi_{0}(h)\|<\|\lambda_{G}(h)\|$

holds (that is, Inequality (1) doesn’t hold). To prove it, we first recall the
following notation from [DIXMIER, p. 282]: we denote by $\Delta$ the modular
function of $G$, and for all continuous, compactly-supported complex-valued
$f$ on $G$, we denote by $f^{*}$ the map defined by

$\forall g\in G,\quad f^{*}(g)=\Delta(g^{-1})\overline{f(g^{-1})}.$

This is the involution of the involutive algebra $C_{c}(G)$. Now, take $f$ as
in Margulis’ theorem, and let

$h:=\frac{1}{\int_{G}f^{*}*f\mathop{}\\!\mathrm{d}\mu_{G}}f^{*}*f.$

Then $h$ satisfies the aforementioned properties. Moreover, from Kesten’s
theorem, since $G$ is amenable, the operator $\lambda(h)$ has norm $1$; and,
according to Margulis’ theorem, $\pi_{0}(h)$ has norm strictly less than $1$.

###  7.2. An action on a finite-volume homogeneous space with optimal
discrepancy

In this section, we give an example of an action of a non-amenable locally
compact group that acts continuously on a measured topological space by
preserving the measure, where the discrepancy inequality is an equality.

Let
$G:=\operatorname{SL}_{3}(\mathbb{R})\times\operatorname{SL}_{3}(\mathbb{R})$,
and consider the subgroup

$\left\\{\begin{pmatrix}A&v\\\ 0&1\\\ \end{pmatrix}\ |\
A\in\operatorname{SL}_{2}(\mathbb{R}),\
v\in\mathbb{R}^{2}\right\\}\simeq\mathbb{R}^{2}\rtimes\operatorname{SL}_{2}(\mathbb{R})$

of $\operatorname{SL}_{3}(\mathbb{R})$ and consider its embedding $H$ in the
first factor of $G$.

Consider now the unique (unital) ring morphism
$\sigma:\mathbb{Z}[\sqrt{2}]\rightarrow\mathbb{C}$ sending $\sqrt{2}$ to
$-\sqrt{2}$. For a matrix
$A:=(a_{ij})_{i,j=1,2,3}\in\operatorname{SL}_{3}(\mathbb{Z}[\sqrt{2}])$, let
us denote also by $\sigma(A)$ the matrix $(\sigma(a_{ij}))_{i,j=1,2,3}$. Since
$\sigma$ is a ring morphism,

$\sigma(A)\in\operatorname{SL}_{3}(\mathbb{Z}[\sqrt{2}]).$

Consider now the map

$\begin{array}[]{rcl}j:\operatorname{SL}_{3}(\mathbb{Z}[\sqrt{2}])&\rightarrow&G\\\
A&\mapsto&(A,\sigma(A)).\\\ \end{array}$

Then $j$ is an injective group morphism, and
$\Gamma:=j(\operatorname{SL}_{3}(\mathbb{Z}[\sqrt{2}]))$ is an irreducible
lattice in $G$ (see [ZIMMER, Example 2.2.5, p.18]). Obviously, as $H$ is
contained in the first factor of $G$, it follows that $\Gamma\cap H=\\{e\\}$.

######  Proposition 5.

Consider the action $H\curvearrowright(G/\Gamma,\mu_{G/\Gamma})$ and the sub-
representation $\pi_{0}$ of the Koopman representation of $H$ on
$L^{2}_{0}(G/\Gamma,\mu_{G/\Gamma})$.

Then, for every positive finite Borel regular measure $\mu$ on $H$, the
inequality in Corollary 1 is optimal, that is, we have

$\|\lambda_{H}(\mu)\|=\|\pi_{0}(\mu)\|.$

###### Proof.

By construction, we can apply Corollary 1, so we have,
$\|\pi_{0}(\mu)\|\geq\|\lambda_{H}(\mu)\|$. For the other inequality, notice
that since $\Gamma$ is irreducible, the action $G\curvearrowright G/\Gamma$ is
mixing (see for example [pinohowemoore, Corollary 3]), and therefore the
restriction of this action to any non-compact subgroup is ergodic. In
particular, the restriction of $\pi_{0}$ to the embedded copy of
$\mathbb{R}^{2}$ in $H$ has no nontrivial invariant vectors. So, according to
[ZIMMER, Theorem 7.3.9, p.146]111The reader willing to read in Zimmer’s book
the result we quote should be warned that in the literature, the symbol
$\prec$ can denote two slightly different notions of weak containment (see
[BHV, Remark F.1.2 (ix), p. 397] for a detailed account of the differences
between the two)., we have the weak containment

$\pi_{0}\prec\lambda_{H}.$

According to [BekDelaHa, Remark 8.B.6 (2) page 244]) this implies the
inequality $\|\pi_{0}(\mu)\|\leq\|\lambda_{H}(\mu)\|$. ∎

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