Let us now assume that $2\le q<\infty$. As before, by Theorem 4.1, the mild
solution $u$ to the problem (5.9) has a continuous modification with values in
$H^{\lambda}({\mathbb{R}})$ which satisfies

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u_{t}|_{H^{\lambda,q}({\mathbb{R}})}^{p}\le
C_{p,q}^{p}\mathbb{E}\|g\|_{L^{2}(0,T;H^{\lambda,q}({\mathbb{R}};\ell^{2}))}^{p},$

where may take $C_{p,q}=10\sqrt{p}(q-1)$ if $2\le p,\infty$. As before, for
$\beta\ge 0$ let

$E_{n,\beta}:=\Big{(}\mathbb{E}\sup_{j=0,\ldots,n}\|u_{t_{j}^{(n)}}-u_{j}^{(n)}\|^{p}_{H^{\lambda-\beta,q}({\mathbb{R}})}\Big{)}^{1/p}.$

By Theorem 5.14 we have $E_{n,0}\to 0$ for (S) and (IE), and if $q=2$ the same
holds for (CN) by Proposition 5.12.

Table 2 gives the estimates for the errors $E_{n,\beta}$ for suitable
intervals for $\beta$ (up to constants depending on $p,q$). The assertions
follow from Example 5.10 (using Proposition 5.12 for (CN) if $q=2$), Corollary
5.2, and Theorem 5.13 applied with $X=H^{\lambda-\beta,q}({\mathbb{R}})$,
$\mathsf{D}(A^{m})=H^{\lambda-\beta+m,q}({\mathbb{R}})$ and
$Y=H^{\lambda,q}({\mathbb{R}})=[X,\mathsf{D}(A^{m})]_{\beta/m}$ for $m=1$ for
(S), $m=2$ for (IE), and $m=3$ for (CN). Note that $\phi(8/5)=1$; since the
convergence rate cannot exceed 1, there is no point in considering values
$\beta>\frac{8}{5}$.

Scheme | $\beta$ | $q$ | Error $E_{n,\beta}$
---|---|---|---
splitting | $(0,1]$ | $[2,\infty)$ | $n^{-\beta}$
implicit Euler | $(0,2]$ | $[2,\infty)$ | $n^{-\beta/2}(\log(n+1))^{1/2}$
Crank-Nicholson | $(0,\frac{3}{2}]$ | $q=2$ | $n^{-2\beta/3}(\log(n+1))^{1/2}$
Crank-Nicholson | $(\frac{1}{2},\frac{8}{5}]$ | $q\in(2,\infty)$ | $n^{-\phi(\beta)}(\log(n+1))^{1/2}$
Table 2. Approximation errors for the stochastic transport equation, where
$\phi$ is the piecewise linear function connecting the points
$(\frac{1}{2},0)$, $(1,\frac{1}{2})$, and $(2,\frac{4}{3})$.

Our final example concerns the Schrödinger equation.

######  Example 5.17 (Stochastic Schrödinger equation).

Consider the following heat equation on ${\mathbb{R}}^{d}$:

$\begin{cases}\,{\rm d}u_{t}(x)&=i\Delta u_{t}+\sum_{k\ge 1}g_{t}^{k}\,{\rm
d}W_{t}^{k},\quad t\in[0,T].\\\ u_{0}&=0.\end{cases}$

We assume that $g\in
L^{p}_{\mathscr{P}}(\Omega;L^{2}(0,T;H^{\lambda}({\mathbb{R}}^{d};\ell^{2})))$
for some $0<p<\infty$, where
$H^{\lambda}({\mathbb{R}}^{d})=H^{\lambda,2}({\mathbb{R}}^{d})$. It is well
known that $i\Delta$ generates a unitary $C_{0}$-group on
$H^{\lambda}({\mathbb{R}}^{d})$ for all $\lambda\in{\mathbb{R}}$. As before,
by Theorem 4.1, the mild solution $u$ to the problem (5.9) has a continuous
modification with values in $H^{\lambda}({\mathbb{R}}^{d})$ which satisfies

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u_{t}\|_{H^{\lambda}({\mathbb{R}}^{d})}^{p}\le
C_{p}^{p}\mathbb{E}\|g\|_{L^{2}(0,T;H^{\lambda}({\mathbb{R}}^{d};\ell^{2}))}^{p},$

where we may take $C_{p}=10\sqrt{p}$ if $2\le p<\infty$.

As before let

$E_{n,\beta}:=\Big{(}\mathbb{E}\sup_{j=0,\ldots,n}\|u_{t_{j}^{(n)}}-u_{j}^{(n)}\|^{p}_{H^{\lambda-2\beta}({\mathbb{R}}^{d})}\Big{)}^{1/p}.$

By Theorem 5.14, $E_{n,0}\to 0$ for (S), (IE), and (CN) (using Proposition
5.12 for the latter).

Table 3 gives the estimates for the errors $E_{n,\beta}$ (up to constants
depending on $p$) for suitable intervals for $\beta$. The assertions follow
from Example 5.10, Corollary 5.2, and Theorem 5.13 applied with
$X=H^{\lambda-2\beta}({\mathbb{R}}^{d})$ and
$Y=H^{\lambda,q}({\mathbb{R}}^{d})=[X,\mathsf{D}(A^{m})]_{\beta/m}$ for $m=1$
for (S), $m=2$ for (IE), and $m=3$ for (CN).

Scheme | $\beta$ | Error $E_{n,\beta}$
---|---|---
splitting | $(0,1]$ | $n^{-\beta}$
implicit Euler | $(0,2]$ | $n^{-\beta/2}(\log(n+1))^{1/2}$
Crank-Nicholson | $(0,\frac{3}{2}]$ | $n^{-2\beta/3}(\log(n+1))^{1/2}$
Table 3. Approximation errors for the stochastic Schrödinger equation.

We are aware of only few papers dealing with convergence uniformly in time in
infinite dimensions. In [GK03] the splitting method is considered for
(possibly degenerate) parabolic problems with gradient noise. The
inhomogeneities have to be uniformly bounded in time. The same methods are
considered in [CvN13] for semi-linear stochastic parabolic problems. No
contractivity of the semigroups needs to be assumed and convergence in Hölder
norms is obtained under $L^{p}$-integrability conditions in time with $p>2$.

In [CvN13] (in the setting of UMD spaces) and [GM07] (in the setting of
monotone operators on Gelfand triples $V\hookrightarrow X\hookrightarrow
V^{*}$), the implicit Euler scheme was considered with uniform convergence in
time, but these results seem not to be comparable to ours due to the fact that
an additional discretisation of the noise term is allowed. In the latter
reference, convergence rates of order $n^{-\nu}$ are obtained under the
assumption that the solution $u$ belong to $C^{\nu}([0,T];L^{2}(\Omega;V))\cap
L^{2}(\Omega;L^{\infty}(0,T;V))$. Results on uniform convergence in time (and
sometimes even convergence in Hölder norms in time) for schemes involving
space and time discretisation can be found in many papers, including [CH12,
CH13, CHJ+16, Gyö99, GM09, Yoo00, Jen09, PS05]. Results concerning uniform
convergence in case of white noise and discretisation in time only can be
found in [BCH19, BG18, GN95, GN97]. Some results are with explicit rates and
some are not, but the schemes considered in these papers are different.

In the parabolic setting, results on convergence of the form

(5.10) $\sup_{j=0,\ldots,n}\mathbb{E}\|u(t_{j}^{(n)})-u_{j}^{(n)}\|^{p}\to 0$

(notice the reversed order of supremum and expectation) with explicit rates,
which can even be faster than $1/n$, can be found in [CvN10, JK11, LPS14] and
references therein.

paper | Scheme | $\beta$ |  $g\in L^{r}$ in time | Error $E_{n}$
---|---|---|---|---
present | splitting | $(0,1]$ | $r=2$ | $n^{-\beta}$
[GK03] | splitting | $2$ | $r=\infty$ | $n^{-1}$
[CvN13] | splitting | $(-\frac{1}{2},\frac{1}{2})$ | $r>2$ | $n^{-\frac{1}{2}-\beta+\frac{1}{r}+\varepsilon}$
Table 4. Comparison of rates in the parabolic setting.

For non-parabolic problems no systematic results seem to be available on
uniform convergence in time. In [Wan15] uniform convergence with explicit
rates has been obtained for a nonlinear wave equation with the splitting
scheme. The fact that the underlying semigroup is a group allows us to write

$\int_{0}^{t}S(t-s)g_{s}dW_{s}=S(t)\int_{0}^{t}S(-s)g_{s}dW_{s}$

and uniform convergence can be obtained from standard maximal estimates for
martingales. In [FTT10] the authors obtain uniform convergence results in case
the semigroup admits a dilation to a group. Our results do not rely on the
above identity and therefore are applicable in the case of arbitrary
contractive $C_{0}$-semigroups, and the convergence holds with the same rate.
Even more is true: for arbitrary $C_{0}$-semigroups and general numerical
schemes the same convergence rates can be obtained up to a logarithmic factor.

##  6\. Maximal inequalities for random stochastic convolutions

In this section we consider the time-dependent problem

(6.1) $\begin{cases}\,{\rm d}u_{t}&=A(t)u_{t}\,{\rm d}t+g_{t}\,{\rm
d}W_{t},\qquad t\in[0,T],\\\ u_{0}&=0.,\end{cases}$

with random operators $A(t)$. More precisely we assume that
$(A(t,\omega))_{(t,\omega)\in[0,T]\times\Omega}$ is an adapted family of
closed operators acting in $X$ which satisfy suitable conditions, to be made
precise below, guaranteeing the generation of an adapted evolution family. We
will assume throughout that $W$ is an adapted $H$-cylindrical Brownian motion
on $\Omega$. and that $g:[0,T]\times\Omega\to\gamma(H,X)$ is progressively
measurable; recall that this is equivalent to the requirement that
$g(h):[0,T]\times\Omega\to X$ is progressively measurable for all $h\in H$.
Many of the results of this section are expected to extend to more general
martingales.

###  6.1. The forward stochastic integral

In analogy with the non-random case one expects that (6.1) admits a mild
solution given as before by the stochastic convolution process
$\int_{0}^{t}S(t,s)g_{s}\,{\rm d}W_{s}$. This stochastic integral, however,
cannot be defined as an Itô stochastic integral because the random variables
$S(t,s)x$ are only assumed to be $\mathscr{F}_{t}$-measurable rather than
$\mathscr{F}_{s}$-measurable and consequently the integrand will not be
progressively measurable in general.

To overcome this problem we use the forward stochastic integral, introduced
and studied by Russo and Vallois [RV93] in the scalar-valued setting.
Following [LN98, PV14, PV15] we define its vector-valued analogue as follows.
Fix an orthonormal basis $(h_{k})_{k\ge 1}$ of $H$. For processes $\Phi\in
L^{0}(\Omega;L^{2}(0,T;\gamma(H,X)))$ and $n=1,2,\dots$ define

$\displaystyle
I^{-}(\Phi,n):=n\sum_{k=1}^{n}\int_{0}^{T}\Phi_{s}h_{k}(W_{(s+1)/n}-W_{s})h_{k}\,{\rm
d}s.$

The process $\Phi$ is forward stochastically integrable if the sequence
$(I^{-}(\Phi,n))_{n\ge 1}$ converges in probability. If this is the case, the
limit is independent of the choice of orthonormal basis and is called the
forward stochastic integral of $\Phi$. We write

$\int_{0}^{T}\Phi_{s}\,{\rm
d}W_{s}^{-}:=I^{-}(\Phi):=\lim_{n\to\infty}I^{-}(\Phi,n).$

Notice that $\Phi$ is not assumed to be progressively measurable. It is easy
to see that if $\Phi$ is a finite rank step process, then $\Phi$ is forward
integrable. In case $\Phi$ is progressively measurable and integrable in the
Itô sense, then the forward stochastic integral exists and coincides with the
Itô integral (see [PV15, Proposition 3.2]).

In order to apply the forward integral to our problem we make following
Hypothesis:

######  Hypothesis 6.1.

The family $(S(t,s,\omega))_{0\le s\le t\le T,\,\omega\in\Omega}$ is an
adapted $C_{0}$-evolution family of contractions on $X$, i.e.,

  1. (i)

$(S(t,s,\omega)_{0\le s\le t\le T}$ is a $C_{0}$-evolution family of
contractions for every $\omega\in\Omega$;

  2. (ii)

$S(t,s,\cdot)x$ is strongly $\mathscr{F}_{t}$-measurable for all $0\le s\le
t\le T$ and $x\in X$.

Furthermore we assume:

  1. (iii)

$Y$ is a Banach space, continuously embedded in $X$, and for almost all
$\omega\in\Omega$ we have $S(t,\cdot,\omega)y\in W^{1,1}(0,t;X)$ for all
$t\in(0,T]$ and $y\in Y$ and

$\|(S(t,\cdot,\omega)y)\|_{W^{1,1}(0,t;X)}\le C(\omega)\|y\|_{Y}$

for some function $C:\Omega\to[0,\infty)$ independent of $y\in Y$ and $0\le
s\le t\le T$.

We have the following sufficient condition for forward integrability (see
[PV15, Corollary 5.3], which extends to the current setting).

######  Proposition 6.2.

Suppose that Hypothesis 6.1 holds, with $X$ a $2$-smooth Banach space, and let
$g:[0,T]\times\Omega\to\gamma(H,Y)$ be a finite rank adapted step process.
Then process $(S(t,s)g_{s})_{s\in[0,t]}$ is forward integrable on $[0,t]$ and
almost surely we have

(6.2) $\displaystyle\int_{0}^{t}S(t,s)g_{s}\,{\rm
d}W_{s}^{-}=S(t,0)\int_{0}^{t}g_{s}\,{\rm
d}W_{s}+\int_{0}^{t}\partial_{s}S(t,s)\int_{s}^{t}g_{r}\,{\rm d}W_{r}\,{\rm
d}s.$

Moreover, the process $(\int_{0}^{t}S(t,s)g_{s}\,{\rm
d}W_{s}^{-})_{t\in[0,T]}$ has a continuous modification.

The right-hand side of (6.2) is well defined by the hypothesis and the
assumption that $g$ takes values in $Y$. By the almost sure pathwise
continuity of $\int_{0}^{\cdot}g_{s}\,{\rm d}W_{s}$, the forward integral in
(6.2) admits a continuous modification.

######  Remark 6.3.

In the setting where $S$ is generated by an adapted family
$(A(t))_{t\in[0,T]}$ satisfying suitable parabolicity assumptions, the right-
hand side of (6.2) is called the pathwise mild solution of (6.1). Pathwise
mild solutions were introduced and extensively studied in [PV14]. In the
parabolic case, $\partial_{s}S(t,s)$ typically extends to a bounded operator
on $X$ and $\|\partial_{s}S(t,s)\|\le C(t-s)^{-1}$, where $C$ depends on
$\omega\in\Omega$. Since $\int_{0}^{\cdot}g_{r}\,{\rm d}W_{r}$ is almost
surely Hölder continuous under $L^{p}(0,T)$-integrability assumptions on $g$
with $p>2$, the right-hand side of (6.2) exists pathwise as a Bochner
integral.

It is quite difficult to prove estimates for the forward integral directly. A
major advantage of using the right-hand side of (6.2) is that one can obtain
estimates using only Itô and Bochner integrals.

###  6.2. The maximal inequality

We will now extend the maximal estimate of Theorem 4.1 to random evolution
families, replacing the Itô stochastic integral of that theorem by the forward
stochastic integral. The precise sense in which the forward integral
constitutes a solution of the problem (6.1) will be addressed subsequently in
Theorem 6.6. Even without the supremum on the left-hand side, the estimate in
Theorem 6.4 is new.

######  Theorem 6.4.

Suppose that Hypothesis 6.1 holds,with $X$ a $2$-smooth Banach space, and let
$g:[0,T]\times\Omega\to\gamma(H,Y)$ be a finite rank adapted step process.
Then for all $0<p<\infty$ we have

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\Big{\|}\int_{0}^{t}S(t,s)g_{s}\,{\rm
d}W_{s}^{-}\Big{\|}^{p}\le
C_{p,D}^{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}^{p},$

where the constant $C_{p,D}$ only depends on $p$ and $D$. For $2\le p<\infty$
the inequality holds with $C_{p,D}=10D\sqrt{p}$.

###### Proof.

The proof is similar to that of Theorem 4.1, but with some extra
technicalities which justify a detailed presentation.

Step 1. Let $g:[0,T]\times\Omega\to\gamma(H,X)$ be an adapted finite rank step
process, say

$g=\sum_{j=1}^{k}{\bf
1}_{(s_{j-1},s_{j}]}\sum_{i=1}^{\ell}h_{i}\otimes\xi_{ij}$

as in (2.8). For the moment there is no need to insist that $g$ be $Y$-valued;
this will only be needed in the last step of the proof.

Fix $0<\delta<T$ and set $S^{\delta}(t,s):=S((t-\delta)^{+},(s-\delta)^{+})$
for $0\le s\le t\le T$. Fix a partition $\pi:=\\{r_{0},\dots,r_{N}\\}$, where
$0=r_{0}<r_{1}<\ldots<r_{N}=T$, and let $(K(t,s,\omega))_{0\le s\le t\le
T,\,\omega\in\Omega}$ be a family of contractions on $X$ with the following
properties:

  1. (i)

$K(t,\cdot,\omega)$ is constant on $[r_{j-1},r_{j})$ for all $t\in[0,T]$,
$\omega\in\Omega$, and $j=1,\ldots,N$;

  2. (ii)

$K(\cdot,s,\omega)$ is strongly continuous for all $s\in[0,T]$ and
$\omega\in\Omega$;

  3. (iii)

$S^{\delta}(t,r,\omega)K(r,s,\omega)=K(t,s,\omega)$ for all $0\le r\le s\le
t\le T$ and $\omega\in\Omega$;

  4. (iv)

$K(t,s,\cdot)x$ is strongly $\mathscr{F}_{(t-\delta)^{+}}$-measurable for all
$0\le s\le t\le T$.

By refining $\pi$ we may assume that $|r_{j}-r_{j-1}|\le\delta$ for
$j=1,\ldots,N$ and that $s_{j}\in\pi$ for all $j=0,\ldots,k$.

Define the process $(v_{t})_{t\in[0,T]}$ by

(6.3) $\displaystyle v_{t}:=\int_{0}^{t}K(t,s)g_{s}\,{\rm d}W_{s}^{-},$

this forward integral being well defined since the integrand is a finite rank
step process. For $t\in[0,r_{1}]$ the above integral coincides with the Itô
integral since $K(t,s,\cdot)$ is strongly $\mathscr{F}_{0}$-measurable. By
(iii), for $r_{j-1}\le s\le t<r_{j}$ we have

(6.4) $\displaystyle v_{t}=S^{\delta}(t,s)v_{s}+\int_{s}^{t}K(t,r)g_{r}\,{\rm
d}W_{r},$

where the stochastic integral is again an Itô integral since the random
variable $K(t,r,\cdot)$ does not depend on $r\in[s,t]\subseteq[r_{j-1},r_{j})$
by (i) and is strongly $\mathscr{F}_{r_{j-1}}$-measurable by (iv) and the
inclusion $\mathscr{F}_{(t-\delta)^{+}}\subseteq\mathscr{F}_{r_{j-1}}$ (using
that $(t-\delta)^{+}\le r_{j-1}$). Properties (i) and (ii) imply that $v$ has
a modification with continuous paths. Working with such a modification, we
will first prove that for all $2\le p<\infty$ one has

$\displaystyle\Big{\|}\sup_{t\in[0,T]}\|v_{t}\|\Big{\|}_{p}\le
10D\sqrt{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}.$

By a limiting argument it suffices to consider exponents $2<p<\infty$.

Let $\pi^{\prime}=\\{t_{0},t_{1},\ldots,t_{m}\\}\subseteq[0,T]$ be another
partition. It suffices to prove

(6.5) $\displaystyle\Big{\|}\sup_{t\in\pi^{\prime}}\|v_{t}\|\Big{\|}\le
a_{\pi}+10D\sqrt{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}$

with $a_{\pi}=o(\hbox{mesh}(\pi))$ as mesh$(\pi)\to 0$. Refining
$\pi^{\prime}$ if necessary, we may assume that $\pi^{\prime}\subseteq\pi$ and
that mesh$(\pi^{\prime})<\delta$.

For fixed $j=1,\ldots,m$ we have, by (6.4),

$\displaystyle f_{j}:=v_{t_{j}}$
$\displaystyle=S^{\delta}(t_{j},t_{j-1})v_{t_{j-1}}+\int_{t_{j-1}}^{t_{j}}K(t_{j},s)g_{s}\,{\rm
d}W_{s}$ $\displaystyle=:V_{j}f_{j-1}+dG_{j},$

where we set $V_{j}:=S^{\delta}(t_{j},t_{j-1})$ and
$G_{j}:=\int_{0}^{t_{j}}K(t_{j},s)g_{s}\,{\rm d}W_{s}$. We further set
$f_{0}:=0$ and $G_{0}:=0$. As in the proof of Theorem 4.1 the sequence
$(dG_{j})_{j=1}^{m}$ is conditionally symmetric and an application of Theorem
3.1 gives

$\displaystyle\|f^{\star}\|_{p}\le
5p\|dG^{\star}\|_{p}+10D\sqrt{p}\|s(G)\|_{p}.$

Proceeding as in Step 1b of the proof of Theorem 4.1 we obtain (6.5).

Step 2. Fix $n\in{\mathbb{N}}$ and set $\sigma_{n}(s):=j2^{-n}T$ for
$s\in[j2^{-n}T,(j+1)2^{-n}T)$. Set
$S_{n}^{\delta}(t,s):=S((t-\delta)^{+},\sigma_{n}((s-\delta)^{+}))$ and define
$v^{(n)}_{t}$ as in (6.3) with $K(t,s)=S_{n}^{\delta}(t,s)$. The assumptions
(i)– (iv) in Step 1 apply to $K(t,s)=S_{n}^{\delta}(t,s)$, $N=2^{n}$, and
$r_{j}=j2^{-n}T$. By what has been shown in this step, the process $v_{t}$ has
a continuous modification. Moreover, for $n\ge m$ the process

$v^{(n)}_{t}-v^{(m)}_{t}=S^{\delta}(t,s)(v_{s}^{(n)}-v_{s}^{(m)})+\int_{s}^{t}K(t,r)(I-S(\sigma_{n}(r),\sigma_{m}(r)))g_{r}\,{\rm
d}W_{r}$

is strongly progressively measurable. Moreover,

$\displaystyle\Big{\|}\sup_{t\in[0,T]}\|v^{(n)}-v^{(m)}\|\Big{\|}_{p}$
$\displaystyle\qquad\qquad\le
C\big{\|}(I-S(\sigma_{n}((\cdot-\delta)^{+}),\sigma_{m}((\cdot-\delta)^{+})))g\big{\|}_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}.$

Since the right-hand side tends to $0$ by dominated convergence,
$(v^{(n)})_{n\ge 1}$ is a Cauchy sequence with respect to the norm of
$L^{p}(\Omega;C([0,T];X))$ and hence converges to some
$\widetilde{v}^{\delta}\in L^{p}(\Omega;C([0,T];X))$. By Step 1,

(6.6)
$\displaystyle\Big{\|}\sup_{t\in[0,T]}\|\widetilde{v}_{t}^{\delta}\|\Big{\|}_{p}=\lim_{n\to\infty}\Big{\|}\sup_{t\in[0,T]}\|v^{(n)}_{t}\|\Big{\|}_{p}\le
C\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}.$

We will show next that
$\widetilde{v}_{t}^{\delta}=\int_{0}^{t}S^{\delta}(t,s)g_{s}\,{\rm
d}W_{s}^{-}$ almost surely for each $t\in[0,T]$. To this end let
$\pi^{\prime\prime}=\\{t_{0},\ldots,t_{M}\\}$ with $0=t_{0}<\ldots<t_{M}=T$
with mesh$(\pi^{\prime\prime})<\delta$. We define an $X$-valued process
$(v_{t}^{\delta})_{t\in[0,T]}$ by setting $v^{\delta}_{0}:=0$ and,
recursively,

$v_{t}^{\delta}=S^{\delta}(t,t_{j-1})v_{t_{j-1}}+\int_{t_{j-1}}^{t}S^{\delta}(t,s)g_{s}\,{\rm
d}W_{s},\qquad t\in(t_{j-1},t_{j}].$

The stochastic integral is well defined since for all $t_{j-1}\le s\le t\le
t_{j}$ the random variable $S^{\delta}(t,s)=S((t-\delta)^{+},(s-\delta)^{+})$
is strongly $\mathscr{F}_{t_{j-1}}$-measurable. Using the elementary
properties of forward integrals we can rewrite this definition as the forward
integral

(6.7) $\displaystyle v^{\delta}(t)=\int_{0}^{t}S^{\delta}(t,s)g_{s}\,{\rm
d}W_{s}^{-},\qquad t\in[0,T].$

We claim that for each $t\in[0,T]$ we have
$v^{\delta}(t)=\widetilde{v}^{\delta}(t)$ almost surely. Indeed, by (2.9),

$\displaystyle\Big{\|}\int_{t_{j-1}}^{t}S^{\delta}_{n}(t,s)g_{s}$
$\displaystyle\,{\rm d}W_{s}-\int_{t_{j-1}}^{t}S^{\delta}(t,s)g_{s}\,{\rm
d}W_{s}\Big{\|}_{L^{2}(\Omega;X)}$ $\displaystyle\le
D\|(S^{\delta}_{n}(t,s)-S^{\delta}(t,s))g_{s}\|_{L^{2}(\Omega;L^{2}(0,t;\gamma(H,X)))}\to
0$

as $n\to\infty$ by dominated convergence. Therefore, the terms in the
recursive identities (6.7) converge to the correct limit and the claim is
proved.

Step 3. We will next show that

$\lim_{\delta\downarrow 0}\int_{0}^{t}S^{\delta}(t,s)g_{s}\,{\rm
d}W_{s}^{-}=\int_{0}^{t}S(t,s)g_{s}\,{\rm d}W_{s}^{-}$

in $L^{0}(\Omega;C([0,T];X))$. This will be done by providing an alternative
formula for $\int_{0}^{t}S^{\delta}(t,s)g_{s}\,{\rm d}W_{s}^{-}$ in which we
can let $\delta\downarrow 0$. Here it will be important that $g$ takes values
in $Y$.

Fix $t\in(0,T]$. Since $\|\partial_{s}(S(t,s)y)\|_{X}\le C\|y\|_{Y}$ with a
constant $C$ independent of $0<s<t\le T$, it follows from Proposition 6.2 that
the forward stochastic convolution integral
$u_{t}:=\int_{0}^{t}S(t,s)g_{s}\,{\rm d}W_{s}^{-}$ exists and is almost surely
equal to

$S(t,0)\int_{0}^{t}g_{r}\,{\rm
d}W_{r}+\int_{0}^{t}\partial_{s}S(t,s)\int_{s}^{t}g_{r}\,{\rm d}W_{r}\,{\rm
d}s.$

Similarly,

$\displaystyle v^{\delta}(t)$
$\displaystyle=S((t-\delta)^{+},0)\int_{0}^{t}g_{r}\,{\rm
d}W_{r}+\int_{0}^{t}\partial_{s}S((t-\delta)^{+},(s-\delta)^{+})\int_{s}^{t}g_{r}\,{\rm
d}W_{r}\,{\rm d}s$ $\displaystyle=S((t-\delta)^{+},0)\int_{0}^{t}g_{r}\,{\rm
d}W_{r}+\int_{0}^{(t-\delta)^{+}}\partial_{s}S((t-\delta)^{+},s)\int_{s+\delta}^{t}g_{r}\,{\rm
d}W_{r}\,{\rm d}s.$

Letting $\delta\downarrow 0$, by the piecewise strong continuity of
$t\mapsto\partial_{s}S(t,s)$ on $Y$ and dominated convergence we obtain that
$v^{\delta}(t)\to u(t)$ almost surely.

By dominated convergence one also obtains that $u$ has a continuous
modification. To prove the maximal estimate for this modification it suffices
to show that for any finite set $\pi\in[0,T]$,

$\displaystyle\Big{\|}\sup_{t\in\pi}\|u_{t}\|\Big{\|}\le
10D\sqrt{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}.$

Using that (6.6) and $v^{\delta}(t)=\widetilde{v}^{\delta}(t)$ for $t\in\pi$,
this follows from Fatou’s lemma:

$\displaystyle\Big{\|}\sup_{t\in\pi}\|u_{t}\|\Big{\|}_{p}\le\liminf_{\delta\downarrow
0}\Big{\|}\sup_{t\in\pi}\|v_{t}^{\delta}\|\Big{\|}\le
10D\sqrt{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}.$

Step 4. The case $0<p<2$ follows again by using Corollary 3.6 instead of
Theorem 3.1, or by an extrapolation argument involving Lenglart’s inequality.
∎

If the embedding $Y\hookrightarrow X$ is dense we can use the maximal
inequality of the theorem to see that for all $0<p<\infty$ the mapping

$g\mapsto\int_{0}^{t}S(t,s)g_{s}\,{\rm d}W_{s}^{-}$

has a unique extension to a continuous linear operator

$J_{p}:L_{\mathscr{P}}^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))\to
L^{p}(\Omega;C([0,T];X)).$

Moreover, by a standard localisation argument, $J$ has a unique extension to a
continuous linear operator

$J:L_{\mathscr{P}}^{0}(\Omega;L^{2}(0,T;\gamma(H,X)))\to
L^{0}(\Omega;C([0,T];X)).$

It is not guaranteed, however, that for general $g\in
L_{\mathscr{P}}^{0}(\Omega;L^{2}(0,T;\gamma(H,X)))$ the process $Jg$ is given
by a forward stochastic convolution again, nor is this clear if we replace
$L^{0}$ and $J$ by $L^{p}$ and $J_{p}$. The same problem occurs if we use the
right-hand side in the identity in Proposition 6.2.

Since $J_{p}$ satisfies the same estimate as in Theorem 6.4, we immediately
obtain an extension of the exponential tail estimate of Corollary 4.2 in the
current setting. As in Remark 4.3 under more restrictive conditions on the
random evolution family, but with better bound on the variance $\sigma^{2}$ a
similar result was obtained in [NV20, Remark 5.8].

The next theorem addresses the question in what sense $J_{p}g$ and $Jg$
“solve” the problem (6.1). Some additional assumptions are needed to establish
the precise relation between the random evolution family $S$ and the random
operator $A$.

######  Hypothesis 6.5.

Hypothesis 6.1 is satisfied. Furthermore, the random operator family
$A:[0,T]\times\Omega\to\mathscr{L}(Y,X)$ has the property that $Ay$ is
strongly progressively measurable for all $y\in Y$. Furthermore the following
conditions hold:

  1. (i)

For almost all $\omega\in\Omega$ we have $S(t,\cdot,\omega)y\in
W^{1,1}(0,t;X)$ for all $t\in[0,T]$ and $y\in Y$, and for almost all
$s\in[0,t]$ we have $\partial_{s}S(t,s)y=-S(t,s)A(s)y$ and

$\|S(t,s)A(s)y\|_{X}\le C\|y\|_{Y},$

where $C:\Omega\to[0,\infty)$ is independent of $y\in Y$ and $0\le s<t\le T$.

  2. (ii)

For almost all $\omega\in\Omega$ we have $S(\cdot,s,\omega)y\in
W^{1,1}(s,T;X)$ for all $s\in[0,T]$ and $y\in Y$, and for almost all
$t\in[s,T]$ we have $\partial_{t}S(t,s)y=A(t)S(t,s)y$ and

$\|A(t)S(t,s)y\|_{X}\le C\|y\|_{Y},$

where $C:\Omega\to[0,\infty)$ is independent of $y\in Y$ and $0\le s<t\le T$.

  3. (iii)

There exists a dense subspace $F\subseteq X^{*}$ such that
$F\subseteq\mathsf{D}(A(t,\omega)^{*})$ for all
$(t,\omega)\in[0,T]\times\Omega$, and almost surely the mapping
$(t,\omega)\mapsto\langle x,A(t,\omega)^{*}x^{*}\rangle$ belongs to
$L^{\infty}(0,T)$ for all $x\in X$ and $x^{*}\in F$.

In the proof below we will combine (iii) with the observation that if
$f:(0,T)\to X$ is integrable and $g:(0,T)\to X^{*}$ has the property that
$\langle x,g\rangle\in L^{\infty}(0,T)$ for all $x\in X$, then the function
$t\mapsto\langle f(t),g(t)\rangle$ is integrable and

$\int_{0}^{T}|\langle f(t),g(t)\rangle|\le\|f\|_{1}\sup_{\|x^{*}\|\le
1}\|\langle x,g\rangle\|_{\infty},$

the supremum on the right-hand side being finite by a closed graph argument.
Indeed, this estimate is clear for simple functions $f$ and the general case
follows by approximation.

Under the above hypothesis a process $u\in
L_{\mathscr{P}}^{0}(\Omega;L^{1}(0,T;X))$ is called a weak solution of (6.1)
if for all $x^{*}\in F$, a.s. for all $t\in[0,T]$,

$\langle u_{t},x^{*}\rangle=\int_{0}^{t}\langle
u_{s},A(s)^{*}x^{*}\rangle\,{\rm d}s+\int_{0}^{t}g_{s}^{*}x^{*}\,{\rm
d}W_{s}.$

In many situations weak solutions are known to be unique. However, we will not
address this issue here.

######  Theorem 6.6.

Suppose that Hypothesis 6.5 holds, with $X$ a $2$-smooth Banach space, and
assume in addition that the embedding $Y\hookrightarrow X$ is dense. Then for
every $g\in L_{\mathscr{P}}^{0}(\Omega;L^{2}(0,T;\gamma(H,X)))$ the process
$Jg$ is a weak solution to (6.1).

###### Proof.

We proceed in three steps.

Step 1. First let $g:[0,T]\times\Omega\to\mathscr{L}(H,Y)$ be an adapted
finite rank step processes and write $v^{g}_{t}=\int_{0}^{t}g\,{\rm d}W$. From
Proposition 6.2, Theorem 6.4 and Hypothesis 6.5(i) it is immediate that

(6.8) $\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u^{g}_{t}\|^{p}\le
C_{p,D}^{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))}^{p},$

and

$u_{t}^{g}=S(t,0)v_{t}^{g}-\int_{0}^{t}S(t,s)A(s)(v_{t}^{g}-v_{s}^{g})\,{\rm
d}s.$

We check next that $u^{g}$ is a weak solution. For this we use a variation of
the argument in [PV14, Theorem 4.9]. For all $x\in Y$,

(6.9) $\displaystyle\int_{0}^{t}S(t,s)A(s)x\,{\rm d}s=-x+S(t,0)x\ \ \
\text{and}\ \ \int_{r}^{t}A(s)S(s,r)x\,{\rm d}r=S(t,r)x-x$

Therefore, applying the first part of (6.9) with $x=v_{t}^{g}$, we obtain

(6.10) $\displaystyle
u_{t}^{g}=v_{t}^{g}+\int_{0}^{t}S(t,s)A(s)v_{s}^{g}\,{\rm d}s.$

To claim that $u^{g}$ is a weak solution it remains to check that

$\left<\int_{0}^{t}S(t,s)A(s)v_{s}^{g}\,{\rm
d}s,x^{*}\right>=\int_{0}^{t}\langle u^{g}_{s},A(s)^{*}x^{*}\rangle\,{\rm
d}s.$

Note that the integral on the right-hand side is well defined as a Lebesgue
integral almost surely. To prove the claim we note that by (6.10), Fubini’s
theorem and the second part of (6.9) (or rather, its weak version
$\int_{r}^{t}\langle S(s,r)x,A^{*}(s)x^{*}\rangle\,{\rm d}r=\langle
S(t,r)x,x^{*}\rangle-\langle x,x^{*}\rangle$, the point being that in the
argument below the vector $x=A(t)v^{g}_{r}$ need not belong to $Y$),

$\displaystyle\int_{0}^{t}\langle$ $\displaystyle
u^{g}_{s},A(s)^{*}x^{*}\rangle\,{\rm d}s$ $\displaystyle=\int_{0}^{t}\langle
v_{s}^{g},A(s)^{*}x^{*}\rangle\,{\rm d}s+\int_{0}^{t}\int_{0}^{s}\langle
S(s,r)A(r)v_{r}^{g},A(s)^{*}x^{*}\rangle\,{\rm d}r\,{\rm d}s$
$\displaystyle=\int_{0}^{t}\langle v_{s}^{g},A(s)^{*}x^{*}\rangle\,{\rm
d}s+\int_{0}^{t}\int_{r}^{t}\langle
S(s,r)A(r)v_{r}^{g},A(s)^{*}x^{*}\rangle\,{\rm d}s\,{\rm d}r$
$\displaystyle=\int_{0}^{t}\langle v_{s}^{g},A(s)^{*}x^{*}\rangle\,{\rm
d}s+\int_{0}^{t}\langle S(t,r)A(r)v_{r}^{g},x^{*}\rangle\,{\rm
d}r-\int_{0}^{t}\langle A(r)v_{r}^{g},x^{*}\rangle\,{\rm d}r$
$\displaystyle=\int_{0}^{t}\langle S(t,r)A(r)v_{r}^{g},x^{*}\rangle\,{\rm
d}r,$

which gives the required identity.

Step 2. Let $g\in L^{p}_{\mathscr{P}}(\Omega;L^{2}(0,T;\gamma(H,X)))$ with
$0<p<\infty$ and choose a sequence of $Y$-valued adapted finite rank step
processes $(g^{(n)})_{n\ge 1}$ such that $g^{(n)}\to g$ in
$L^{p}(\Omega;L^{2}(0,T;\gamma(H,X)))$. Then from (6.8) applied to
$g^{(n)}-g^{m}$ we obtain that $(u^{g^{(n)}})_{n\ge 1}$ is a Cauchy sequence
and therefore converges to some $u$ in $L^{p}(\Omega;C([0,T];X))$. By Step 1,
$u^{g^{(n)}}$ is a weak solution and thus

$\langle u_{t}^{g^{(n)}},x^{*}\rangle=\int_{0}^{t}\langle
u_{s}^{g^{(n)}},A(s)^{*}x^{*}\rangle\,{\rm
d}s+\int_{0}^{t}(g_{s}^{(n)})^{*}x^{*}\,{\rm d}W_{s}.$

Letting $n\to\infty$ in this identity we conclude that $u^{g}$ is a weak
solution. The maximal inequality is obtained by applying (6.8) with $g_{n}$
and letting $n\to\infty$. ∎

######  Remark 6.7.

In [LN98, Proposition 5.3], restrictive conditions in terms of Malliavin
differentiability of $S$ are given under which the forward stochastic integral
$u_{t}=\int_{0}^{t}S(t,s)g_{s}\,{\rm d}W_{s}^{-}$ exists, has a continuous
modification, and is a weak solution. Inspection of the proof shows that that
if one sets $u_{t}^{(n)}:=I^{-}({\bf 1}_{[0,t]}S(t,\cdot)g^{(n)})$, one needs
that $\sup_{t\in[0,T]}\|u_{t}-u_{t}^{(n)}\|_{L^{1}(\Omega;X)}\to 0$. Although
this is likely to hold in many situations, such considerations can be avoided
by using the right-hand side of (6.2).

######  Remark 6.8.

Theorem 5.1 extend mutatis mutandis to random evolution families. The only
required change is to use the forward integral in the proof and to apply
Theorem 6.4 instead of Theorem 4.1. To obtain explicit decay rates under the
assumption that $g$ has spatial smoothness, i.e., $g$ takes values in a Banach
space $Y$ continuously embedded in $X$, one requires estimates for
$\|S(s,\sigma_{n}(s))-I\|_{\mathscr{L}(Y,X)}$. In some applications (e.g.
[Paz83, Section 5.2]) such estimates are available.

Acknowledgment. We thank Antonio Agresti, Sonja Cox, Kristin Kirchner, Emiel
Lorist, and Ivan Yaroslavtsev for helpful comments.

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# Bohr neighborhoods in three-fold difference sets

John T. Griesmer  Department of Applied Mathematics and Statistics
Colorado School of Mines, Golden, Colorado<EMAIL_ADDRESS>
###### Abstract.

Answering a question of Hegyvári and Ruzsa, we show that if $A$ is a set of
integers having positive upper Banach density, then the set
$A+A-A:=\\{a+b-c:a,b,c\in A\\}$ contains Bohr neighborhoods of many elements
of $A$, where the radius and dimension of the Bohr neighborhood depend only on
$d^{*}(A)$.

##  1\. Introduction

For a real number $t$, $\|t\|$ denotes distance to the nearest integer. If
$k\in\mathbb{N}$, $\eta>0$, and $s_{1},\dots,s_{k}$ are real numbers, then a
_Bohr- $(k,\eta)$ set_ is a set of integers of the form
$\\{n:\max_{i\in\\{1,\dots,k\\}}\|s_{i}n\|<\eta\\}$. A Bohr-$(k,\eta)$
neighborhood of $n$ is a set of the form $n+U$, where $U$ is a Bohr-$(k,\eta)$
set. The parameters $\eta$ and $k$ are called the _radius_ and _dimension_ of
$U$, respectively.

Let $d^{*}(A)$ denote the upper Banach density of a set
$A\subseteq\mathbb{Z}$. Theorem 2.2 of [6] says that if $d^{*}(A)>0$, then
$A+A-A$ is a Bohr neighborhood of many $a\in A$. The proof therein does not
specify parameters $k,\eta$ for the Bohr neighborhood in terms of $d^{*}(A)$,
and Section 3 of [6] asks for a proof which makes those parameters effective.
Our main result is the following theorem, which provides the requested
effective bounds.

######  Theorem 1.1.

  1. 1.

Let $A\subseteq\mathbb{Z}$ have $d^{*}(A)>0$. There are constants
$k\in\mathbb{N}$, $\eta>0$, depending only on $d^{*}(A)$, such that $A+A-A-a$
contains a Bohr-$(k,\eta)$ set for some $a\in A$.

  2. 2.

For all $\varepsilon>0$, there are constants $k\in\mathbb{N},\eta>0$,
depending only on $d^{*}(A)$ and $\varepsilon$, such that there is a set
$A^{\prime}\subseteq A$ satisfying $d^{*}(A\setminus A^{\prime})<\varepsilon$,
and $A+A-A-a^{\prime}$ contains a Bohr-$(k,\eta)$ set for all $a^{\prime}\in
A^{\prime}$.

The estimate $d^{*}(A\setminus A^{\prime})<\varepsilon$ cannot be improved to
$d^{*}(A\setminus A^{\prime})=0$, but we omit examples to this effect, as the
constructions are tedious.

Our proof of Theorem 1.1 generalizes without modification to the setting of
countable abelian groups, so we work in that context. The next section
introduces some terminology and notation for countable abelian groups, and
states Theorem 2.2, the natural generalization of Theorem 1.1 to that setting.
The proof of Theorem 2.2 is supplied by Proposition 4.1, which reduces the
study of $A+A-A$ to an analogous problem for compact abelian groups, solved in
Section 3. The proof of Proposition 4.1 is carried out in Section 5 using
ergodic theoretic methods similar to those of [4, 5].

It would be interesting to find a shorter, more elementary proof of Theorem
1.1.

##  2\. Countable abelian groups

Let $\Gamma$ be a countable abelian group. If $A,B\subseteq\Gamma$,
$\gamma\in\Gamma$, write $A+B$ for $\\{a+b:a\in A,b\in B\\}$, $A-B$ for
$\\{a-b:a\in A,b\in B\\}$, and $\gamma+A$ for $\\{\gamma+a:a\in A\\}$.

###  2.1. Upper Banach density

A Følner sequence for $\Gamma$ is a sequence $(\Phi_{n})_{n\in\mathbb{N}}$ of
finite subsets of $\Gamma$ such that
$\lim_{n\to\infty}\frac{|\Phi_{n}\triangle(\Phi_{n}+\gamma)|}{|\Phi_{n}|}=0$
for every $\gamma\in\Gamma$. It is well known that every countable abelian
group admits a Følner sequence. If $\mathbf{\Phi}=(\Phi_{n})_{n\in\mathbb{N}}$
is a Følner sequence and $A\subseteq\Gamma$, let
$\overline{d}_{\mathbf{\Phi}}(A):=\limsup_{n\to\infty}\frac{|A\cap\Phi_{n}|}{|\Phi_{n}|}$
be the _upper density of $A$ with respect to $\mathbf{\Phi}$_. Write
$d_{\mathbf{\Phi}}(A)$ for $\overline{d}_{\mathbf{\Phi}}(A)$ if the limit
exists. For $A\subseteq\Gamma$, the _upper Banach density of $A$_ is
$d^{*}(A):=\sup\\{d_{\mathbf{\Phi}}(A):\mathbf{\Phi}\text{ is a F{\o}lner
sequence}\\}$.

###  2.2. Bohr neighborhoods

Let $\mathbb{T}$ denote the group $\mathbb{R}/\mathbb{Z}$. For
$t\in\mathbb{R}$, recall that $\|t\|$ is the distance from $t$ to the nearest
integer. For $x\in\mathbb{T}$, $\|x\|$ is defined to be $\|\tilde{x}\|$, where
$\tilde{x}\in\mathbb{R}$ satisfies $\tilde{x}+\mathbb{Z}=x$.

If $S$ is a finite set of homomorphisms $\rho:\Gamma\to\mathbb{T}$, $|S|=k$,
and $\eta>0$, then $\\{\gamma\in\Gamma:\|\rho(\gamma)\|<\eta\text{ for all
}\rho\in S\\}$ is a called a _Bohr- $(k,\eta)$ set_. A _Bohr- $(k,\eta)$
neighborhood_ of $a\in\Gamma$ is a set of the form $a+U$, where $U$ is a
Bohr-$(k,\eta)$ set. These definitions of “Bohr neighborhood” and “Bohr set”
agree with the definitions in Section 1, as the maps $n\mapsto
s_{i}n+\mathbb{Z}$ are homomorphisms from $\mathbb{Z}$ to $\mathbb{T}$.

Let $\mathcal{S}^{1}$ denote the group of complex numbers of modulus $1$, with
the group operation of multiplication. The groups $\mathbb{T}$ and
$\mathcal{S}^{1}$ are isomorphic, via the isomorphism
$e_{1}:\mathbb{T}\to\mathcal{S}_{1}$, $e_{1}(t):=\exp(2\pi it)$. This leads to
the following observation.

######  Observation 2.1.

If $S$ is a finite set of homomorphisms $\rho:\Gamma\to\mathcal{S}^{1}$,
$|S|=k$, and $\eta>0$, then the set
$\\{\gamma\in\Gamma:|\rho(\gamma)-1|<\eta\text{ for all }\rho\in S\\}$
contains a Bohr-$(k,\eta/(2\pi))$ set.

######  Theorem 2.2.

Let $\varepsilon,\delta>0$, and let $\Gamma$ be a countable abelian group.

  1. 1.

There are constants $k\in\mathbb{N}$, $\eta>0$, depending only on $\delta$,
such that if $A\subseteq\Gamma$, has $d^{*}(A)\geq\delta$, then $A+A-A-a$
contains a Bohr-$(k,\eta)$ set for some $a\in A$.

  2. 2.

There are constants $k^{\prime}\in\mathbb{N},\eta^{\prime}>0$, depending only
on $\delta$ and $\varepsilon$, such that if $A\subseteq\Gamma$ has
$d^{*}(A)\geq\delta$, there is a set $A^{\prime}\subseteq A$ satisfying
$d^{*}(A\setminus A^{\prime})<\varepsilon$, and $A+A-A-a^{\prime}$ contains a
Bohr-$(k^{\prime},\eta^{\prime})$ set for all $a^{\prime}\in A^{\prime}$.

The constants $k,k^{\prime},\eta,\eta^{\prime}$ do not depend on $\Gamma$.

Part 2 follows immediately from Part 1, while Part 1 will be derived from
Proposition 4.1 and proved in Section 4.

###### Proof of Part 2..

Assume Part 1 holds, and assume, to get a contradiction, that Part 2 fails.
Given a set $A\subseteq\Gamma$, $k\in\mathbb{N}$, $\eta>0$, let

$E(A,k,\eta):=\\{a\in A:A+A-A-a\text{ does not contain a Bohr-$(k,\eta)$
set}\\}.$

Then there are $\delta>0$, $\varepsilon>0$ such that for all $k\in\mathbb{N}$,
$\eta>0$, there exists $A$ having $d^{*}(A)>\delta$, while
$d^{*}(E(A,k,\eta))>\varepsilon$. Setting $E:=E(A,k,\eta)$, we have an
$\varepsilon>0$ such that for all $k\in\mathbb{N},\eta>0$, there is a set $E$
satisfying $d^{*}(E)>\varepsilon$ and for all $e\in E$, $E+E-E-e$ does not
contain a Bohr-$(k,\eta)$ set. This contradicts Part 1. ∎

##  3\. Compact abelian groups

###  3.1. Bohr neighborhoods in topological abelian groups

######  Definition 3.1.

If $Z$ is a topological abelian group, $S$ is a set of _continuous_
homomorphisms $\rho:Z\to\mathbb{T}$, $|S|=k$, and $\eta>0$, then
$\\{z:\|\rho(z)\|<\eta\text{ for all }\rho\in S\\}$ is a _Bohr- $(k,\eta)$
set_. A Bohr-$(k,\eta)$ neighborhood of $z\in Z$ is a set of the form $z+U$,
where $U$ is a Bohr-$(k,\eta)$ set.

If $Z$ is a topological abelian group, $\widehat{Z}$ denotes the character
group of $Z$, meaning the group of continuous homomorphisms
$\chi:Z\to\mathcal{S}^{1}$ with the group operation of pointwise
multiplication.

######  Observation 3.2.

Following Observation 2.1, we see that if $S\subseteq\widehat{Z}$, $|S|=k$,
and $\eta>0$, then the set $\\{z:|\chi(z)-1|<\eta\text{ for all }\chi\in S\\}$
contains a Bohr-$(k,\eta/(2\pi))$ set.

######  Lemma 3.3.

Let $\Gamma$ be a countable abelian group and $Z$ a topological abelian group.
If $\rho:\Gamma\to Z$ is a homomorphism and $U\subseteq Z$ contains a
Bohr-$(k,\eta)$ set, then $\rho^{-1}(U)$ contains a Bohr-$(k,\eta)$ set in
$\Gamma$.

###### Proof.

Let $S$ be a set of $k$ continuous homomorphisms $\psi:Z\to\mathbb{T}$ such
that $U$ contains $\\{z\in Z:\|\psi(z)\|<\eta\text{ for all }\psi\in S\\}$.
Then $\rho^{-1}(U)$ contains
$\\{\gamma\in\Gamma:\|\psi\circ\rho(\gamma)\|<\eta\text{ for all }\psi\in
S\\}$, which is a Bohr-$(k,\eta)$ set. ∎

######  Lemma 3.4.

Let $Z$ be a topological abelian group and $S$ a finite subset of
$\widehat{Z}$. If $p:Z\to\mathbb{C}$, $p:=\sum_{\chi\in S}c_{\chi}\chi$, where
$|c_{\chi}|\leq 1$ for all $\chi$, and $a\in Z$ satisfies $\real p(a)\geq c$,
then $\\{x\in Z:\real p(x)>0\\}$ contains a Bohr-$(k,\eta)$ neighborhood of
$a$, where $k$ and $\eta$ depend only on $|S|$ and $c$. In fact we can use
$k=|S|$ and $\eta=\frac{c}{2\pi|S|}$.

###### Proof.

Let $U:=\\{x:|\chi(x)-1|<\frac{c}{|S|}\text{ for all }\chi\in S\\}$, so that
$U$ is a Bohr-$(|S|,c/(2\pi|S|))$ set, by Observation 3.2. Then for $x\in U$,

$\displaystyle|p(a+x)-p(a)|$ $\displaystyle\leq\sum_{\chi\in
S}|\chi(a+x)-\chi(a)|$ $\displaystyle=\sum_{\chi\in S}|\chi(a)(\chi(x)-1)|$
$\displaystyle=\sum_{\chi\in S}|\chi(x)-1|$ $\displaystyle<c,$

so $\real p(a+x)>0$ for all $x\in U$, and $a+U\subseteq\\{x:\real p(x)>0\\}$.
∎

The following lemma may be proved similarly.

######  Lemma 3.5.

Suppose $U$ is a Bohr-$(k,\eta)$ neighborhood of $a\in Z$. Then there is a
neighborhood $V$ of $a$ (in the topology of $Z$) such that $U$ is a
Bohr-$(k,\eta/2)$ neighborhood of $z$ for all $z\in V$.

###  3.2. Fourier identities

We summarize some of the basic facts and definitions from harmonic analysis on
compact abelian groups, available in standard references such as [7].

Let $Z$ be a compact abelian group with Haar measure $m$, normalized so that
$m(Z)=1$. For $\chi\in\widehat{Z}$ and $f\in L^{2}(m)$, $\hat{f}(\chi):=\int
f\cdot\overline{\chi}\,dm$, and the function
$\hat{f}:\widehat{Z}\to\mathbb{C}$ is the _Fourier transform_ of $f$. For
$f,g\in L^{1}(m)$, define the convolution $f*g$ by

$f*g(z):=\int f(z-t)g(t)\,dm(t).$

Let $f,g\in L^{2}(m)$. Then

(3.1) $\displaystyle\int f\cdot\bar{g}\,dm$
$\displaystyle=\sum_{\chi\in\widehat{Z}}\hat{f}(\chi)\cdot\overline{\hat{g}(\chi)}$
(3.2) $\displaystyle\int|f|^{2}\,dm$
$\displaystyle=\sum_{\chi\in\widehat{Z}}|\hat{f}(\chi)|^{2}$ (3.3)
$\displaystyle\widehat{f*g}$ $\displaystyle=\hat{f}\cdot\hat{g}.$

Let $g_{-}$ be the function defined by $g_{-}(z)=g(-z)$. Then

(3.4) $\widehat{g_{-}}=\overline{\hat{g}}.$

Convolution is associative, so the three-fold convolution $h:=f*g*g_{-}$ is
well-defined. Furthermore, when $f,g\in L^{2}(m)$, $h$ is continuous and its
Fourier series $\sum_{\chi\in\widehat{Z}}\hat{h}(\chi)\cdot\chi$ converges
uniformly to $h$.

######  Lemma 3.6.

Let $\delta>0$. There exist $k\in\mathbb{N}$, $\eta>0$, depending only on
$\delta$, such that if $Z$ is a compact abelian group with Haar measure $m$,
and $f,g:Z\to[0,1]$ are measurable functions having $\int f\,dm,\int
g\,dm\geq\delta$, then $\\{z\in Z:f*g*g_{-}(z)>0\\}$ contains a
Bohr-$(k,\eta)$ neighborhood of some $a\in Z$ having $f(a)>0$.

###### Proof.

Without loss of generality, we assume $\int f\,dm=\int g\,dm$, so let
$f,g:Z\to[0,1]$ have $\int f\,dm=\int g\,dm=\delta$.

Let $h:=f*g*g_{-}$. Then $h:Z\to[0,1]$, $h$ is continuous, and
$\hat{h}=\hat{f}\cdot|\hat{g}|^{2}$, by Equations (3.3) and (3.4).
Consequently $\hat{h}\in l^{1}(\widehat{Z})$. Let
$S_{1}:=\\{\chi\in\widehat{Z}:|\hat{f}(\chi)|\geq\frac{1}{4}\delta^{3}\\}$,
$S_{2}:=\widehat{Z}\setminus S_{1}$. We claim that

(3.5) $|S_{1}|\leq 16\delta^{-5}$

and

(3.6) $h(a)\geq\delta^{4}\text{ for some }a\text{ having }f(a)>0.$

We postpone the proofs of inequalities (3.5) and (3.6) and now prove the
conclusion of the lemma. Write the Fourier series of $h$ as $h(x)=p(x)+r(x)$,
where

$\displaystyle p(x):=\sum_{\chi\in S_{1}}\hat{h}(\chi)\chi(x),\ \
r(x):=\sum_{\chi\in S_{2}}\hat{h}(\chi)\chi(x).$

Both series converge uniformly, since $\hat{h}\in l^{1}(\widehat{Z})$.
Estimating $r(x)$, we get

$\displaystyle\Bigl{|}\sum_{\chi\in S_{2}}\hat{h}(\chi)\chi(x)\Bigr{|}$
$\displaystyle\leq\sum_{\chi\in S_{2}}|\hat{f}(\chi)||\hat{g}(\chi)|^{2}$
$\displaystyle\leq\tfrac{1}{4}\delta^{3}\sum_{\chi\in
S_{2}}|\hat{g}(\chi)|^{2}$ by definition of $S_{2}$
$\displaystyle\leq\tfrac{1}{4}\delta^{3}\sum_{\chi\in\widehat{Z}}|\hat{g}(\chi)|^{2}$
$\displaystyle=\tfrac{1}{4}\delta^{3}\int|g|^{2}\,dm$ by (3.2)
$\displaystyle\leq\tfrac{1}{4}\delta^{3}\int|g|\,dm$ $\displaystyle\text{since
}0\leq g\leq 1$ $\displaystyle\leq\tfrac{1}{4}\delta^{3}\cdot\delta,$

so

(3.7) $|r(x)|\leq\tfrac{1}{4}\delta^{4}\text{ for all }x.$

It follows that $h(x)$ is positive whenever $\real
p(x)>\frac{1}{4}\delta^{4}$, meaning the real part of
$q(x):=p(x)-\frac{1}{4}\delta^{4}$ is positive. Choose an $a_{0}$ so that
$h(a_{0})\geq\delta^{4}$ and $f(a_{0})>0$. Inequality (3.7) implies $\real
p(a_{0})\geq\frac{3}{4}\delta^{4}$, so $\real
q(a_{0})\geq\frac{1}{2}\delta^{4}$. By Lemma 3.4, the set $\\{x:\real
q(x)>0\\}$ contains a Bohr-$(|S_{1}|,c/(2\pi|S_{1}|))$ neighborhood around
$a_{0}$, where $c=\real q(a_{0})\geq\frac{1}{2}\delta^{4}$. This Bohr
neighborhood is contained in $\\{x:h(x)>0\\}$, so we have proved the lemma.

It remains to prove inequalities (3.5) and (3.6). To prove Inequality (3.5),
consider

$\displaystyle|S_{1}|\cdot\tfrac{1}{16}\delta^{6}$
$\displaystyle\leq\sum_{\chi\in S_{1}}|\hat{f}(\chi)|^{2}$
$\displaystyle\leq\sum_{\chi\in\widehat{Z}}|\hat{f}(\chi)|^{2}$
$\displaystyle=\int|f|^{2}\,dm$ by (3.2) $\displaystyle\leq\int|f|\,dm$ since
$0\leq f\leq 1$ $\displaystyle=\delta,$

so $|S_{1}|\leq 16\delta^{-5}$.

To prove Inequality (3.6), consider

$\displaystyle\int h\cdot f\,dm$ $\displaystyle=\int h\cdot\bar{f}\,dm$
$\displaystyle=\sum_{\chi\in\widehat{Z}}\hat{h}(\chi)\overline{\hat{f}(\chi)}$
by (3.1)
$\displaystyle=\sum_{\chi\in\widehat{Z}}|\hat{f}(\chi)|^{2}|\hat{g}(\chi)|^{2}$
$\displaystyle=\sum_{\chi\in\widehat{Z}}|\widehat{f*g}(\chi)|^{2}$ by (3.3)
$\displaystyle=\int|f*g|^{2}\,dm$ by (3.2) $\displaystyle\geq\bigl{|}\int
f*g\,dm\bigr{|}^{2}$ by convexity of $t\mapsto t^{2}$
$\displaystyle=\delta^{4},$

so that $\int h\cdot f\,dm\geq\delta^{4}$. Let $A:=\\{a\in Z:f(a)>0\\}$.
Observe that $\sup_{a\in A}h(a)\geq\int h\cdot f\,dm$, and equality holds only
if $h$ is constant, so we conclude that $h(a)\geq\delta^{4}$ for some $a\in
A$. ∎

##  4\. From countable to compact

######  Proposition 4.1.

Let $\Gamma$ be a countable abelian group and let $A,B\subseteq\Gamma$ have
$d^{*}(A)>0$, $d^{*}(B)>0$. Then there are:

  1. $\bullet$

a compact abelian group $Z$ with normalized Haar measure $m$,

  2. $\bullet$

a homomorphism $\rho:\Gamma\to Z$ such that $\rho(\Gamma)$ is dense in $Z$,

  3. $\bullet$

Borel functions $f,g:Z\to[0,1]$ satisfying $\int f\,dm=d^{*}(A)$, $\int
g\,dm=d^{*}(B)$,

such that

  1. $\bullet$

$f$ is supported on $\overline{\rho(A)}$,

  2. $\bullet$

$\\{\gamma\in\Gamma:f*g*g_{-}(\rho(\gamma))>0\\}\subseteq A+B-B$,

where $\overline{\rho(A)}$ denotes the topological closure of $\rho(A)$ in
$Z$.

We now prove Theorem 2.2 as a consequence of Proposition 4.1 and Lemma 3.6.

###### Proof of Theorem 2.2..

Let $A,B\subseteq\Gamma$ have $d^{*}(A),d^{*}(B)>0$.

Set $\delta:=\min(d^{*}(A),d^{*}(B))$, and let $Z,\rho,f,$ and $g$ be as in
Proposition 4.1. Let

$\displaystyle h$ $\displaystyle:=f*g*g_{-}$ $\displaystyle\tilde{A}$
$\displaystyle:=\\{z\in Z:f(z)>0\\}$ $\displaystyle U$ $\displaystyle:=\\{z\in
Z:h(z)>0\\},$

so that $\tilde{A}\subseteq\overline{\rho(A)}$ and $\rho^{-1}(U)\subseteq
A+B-B$. By Lemma 3.6, there is an $\tilde{a}\in\tilde{A}$ such that
$U-\tilde{a}$ contains a Bohr-$(k,\eta)$ set, where $k,\eta$ depend only on
$\delta$. Lemma 3.5 provides a neighborhood $V$ of $\tilde{a}$ such that $U-z$
contains a Bohr-$(k,\eta/2)$ set for all $z\in V$. Since
$\tilde{a}\in\overline{\rho(A)}$, there is an $a\in A$ such that $\rho(a)\in
V$. For such $a$, Lemma 3.3 implies $\rho^{-1}(U)-a$ is a Bohr-$(k,\eta/2)$
set contained in $A+B-B-a$. ∎

##  5\. Correspondence principle and proof of Proposition 4.1

In this section we fix a countable abelian group $\Gamma$. We will exploit the
theory measure preserving actions of $\Gamma$, see [1], [2], or [3] for
general references, and [4] or [5] for similar applications.

###  5.1. Measure preserving systems

A _measure preserving $\Gamma$-system_ (or briefly, _$\Gamma$ -system_) is a
quadruple $(X,\mathscr{X},\mu,T)$, where $(X,\mathscr{X},\mu)$ is a
probability measure space and $T$ is an action of $\Gamma$ on $X$ preserving
$\mu$:

(5.1) $\mu(T^{-\gamma}D)=\mu(D)$

for all measurable $D\subseteq X$ and all $\gamma\in\Gamma$. Note that
Equation (5.1) yields the identities

(5.2) $\displaystyle\mu(C\cap T^{-\gamma}D)$ $\displaystyle=\mu(T^{a}(C\cap
T^{-\gamma}D))=\mu(T^{a}C\cap T^{a-\gamma}D),$ (5.3) $\displaystyle\int f\cdot
g\circ T^{\gamma}\,d\mu$ $\displaystyle=\int f\circ T^{-\gamma}\cdot g\,d\mu,$

for all measurable $C,D\subseteq X$, all $f,g\in L^{2}(\mu)$, and all
$a,\gamma\in\Gamma$.

A $\Gamma$-system is _ergodic_ if for every $D\subseteq X$ satisfying
$\mu(D\triangle T^{\gamma}D)=0$ for all $\gamma\in\Gamma$, we have $\mu(D)=0$
or $\mu(D)=1$.

A _factor_ of a $\Gamma$-system $(X,\mathscr{X},\mu,T)$ is a $\Gamma$-system
$(Y,\mathscr{Y},\nu,S)$ together with a _factor map_ $\pi:X\to Y$, defined for
$\mu$-a.e. $x\in X$, such that

(5.4) $\pi(T^{\gamma}x)=S^{\gamma}\pi(x)\ \text{ for $\mu$-a.e.\ $x\in X$ and
all $\gamma\in\Gamma$}.$

The space $L^{2}(\nu)$ may be identified with the subspace of $L^{2}(\mu)$
consisting of functions of the form $g\circ\pi$, where $g\in L^{2}(\nu)$. The
$\sigma$-algebra $\pi^{-1}(\mathscr{Y})$ consists of those sets which are
$\mu$-a.e. equal to a set of the form $\pi^{-1}(C)$, where $C\subseteq Y$.
Note that $L^{2}(\nu)$ may be identified with those elements of $L^{2}(\mu)$
which are $\pi^{-1}(\mathscr{Y})$-measurable. Let
$P_{\mathscr{Y}}:L^{2}(\mu)\to L^{2}(\mu)$ denote the orthogonal projection
onto the space of $\pi^{-1}(\mathscr{Y})$-measurable functions.

If $g\in L^{2}(\mu)$, let $\tilde{g}$ be the element of $L^{2}(\nu)$
satisfying $P_{\mathscr{Y}}g=\tilde{g}\circ\pi$. Note that if $f$ is
$\pi^{-1}(\mathscr{Y})$-measurable, we have

(5.5) $\int f\cdot g\circ T^{\gamma}\,d\mu=\int\tilde{f}\cdot\tilde{g}\circ
S^{\gamma}\,d\nu\ \ \text{ for all }\gamma\in\Gamma,$

where $\tilde{f}\in L^{2}(\nu)$ satisfies $f=\tilde{f}\circ\pi$.

###  5.2. Group rotations

A _group rotation_ is a $\Gamma$-system $(Z,\mathcal{Z},m,R_{\rho})$, where
$Z$ is a compact abelian group with normalized Haar measure $m$,
$\rho:\Gamma\to Z$ is a homomorphism, and the action $R_{\rho}$ is given by
$R_{\rho}^{\gamma}(z)=z+\rho(\gamma)$. The group rotation
$(Z,\mathcal{Z},m,R_{\rho})$ is ergodic iff $\rho(\Gamma)$ is dense in $Z$.

The _Kronecker factor_ of a $\Gamma$-system $\mathbf{X}$ is the maximal factor
$\mathbf{Y}$ of $\mathbf{X}$ such that $\mathbf{Y}$ is isomorphic to a group
rotation. When $\mathbf{X}$ is ergodic, such a factor always exists and is
ergodic (although it may be trivial). See [1], [2], or [3] for the existence
of the Kronecker factor and its properties.

###  5.3. A correspondence principle

The following lemma is standard, but we outline the proof for completeness.

######  Lemma 5.1.

If $B\subseteq\Gamma$ has $d^{*}(B)>0$, there is an ergodic $\Gamma$-system
$(X,\mathscr{X},\mu,T)$ and a $D\subseteq X$ having $\mu(D)\geq d^{*}(B)$ such
that $B-B$ contains $R(D):=\\{\gamma:\mu(D\cap T^{\gamma}D)>0\\}$.

###### Proof.

Let $\Omega=\\{0,1\\}^{\Gamma}$ with the product topology, so that $\Omega$ is
a compact metrizable space. Let $T$ be the action of $\Gamma$ on $\Omega$
defined by $(T^{\beta}\omega)(\gamma):=\omega(\gamma+\beta)$. Let
$\mathbf{\Phi}=(\Phi_{n})_{n\in\mathbb{N}}$ be a Følner sequence for $\Gamma$
such that $d_{\mathbf{\Phi}}(B)=d^{*}(B)$. Consider $x:=1_{B}\in\Omega$, and
let $X$ be the orbit closure of $x$ under $T$, meaning $X$ is the closure of
the set $\\{T^{\gamma}x:\gamma\in\Gamma\\}$. Let $D$ be the set $\\{\omega\in
X:\omega(0)=1\\}$, so that $D$ is a clopen subset of $X$. Note that
$B=\\{\gamma:T^{\gamma}x\in D\\}$.

We will find a $T$-invariant measure $\mu$ on $X$ such that $\mu(D)=d^{*}(B)$.
Let $\delta_{x}$ be the Dirac mass concentrated at $x$, and for each $n$, let
$\nu_{n}:=\frac{1}{|\Phi_{n}|}\sum_{\gamma\in\Phi_{n}}\delta_{T^{\gamma}x}$.
Let $\nu$ be a $\text{weak}^{*}$ limit of the $\nu_{n}$. Then $\nu$ is a
$T$-invariant probability measure, while

$\nu(D)=\lim_{n\to\infty}\frac{1}{|\Phi_{n}|}|\\{\gamma\in\Phi_{n}:T^{\gamma}x\in
D\\}|=\lim_{n\to\infty}\frac{|B\cap\Phi_{n}|}{|\Phi_{n}|}=d^{*}(B).$

Applying ergodic decomposition ([1], Theorem 8.20), we may find an ergodic
$T$-invariant measure $\mu$ on $X$ such that $\mu(D)\geq\nu(D)$.

We now show that $\mu(D\cap T^{\gamma}D)>0$ implies $\gamma\in B-B$. In fact,
if $D\cap T^{\gamma}D\neq\varnothing$, then there exists $y\in D$ such that
$T^{\gamma}y\in D$. Since $y$ is a limit of points of the form $T^{\beta}x$
and $D$ is open, there exist $\beta\in\Gamma$ such that $T^{\beta}x\in D$ and
$T^{\gamma}T^{\beta}x\in D$, which implies $\beta\in B$ and $\gamma+\beta\in
B$. We then have $\gamma\in B-B$. ∎

If $D\subseteq X$ and $A\subseteq\Gamma$, let $A\cdot D:=\bigcup_{a\in
A}T^{a}D$.

######  Lemma 5.2.

If $\mathbf{X}$ is the $\Gamma$-system obtained in Lemma 5.1 and
$A\subseteq\Gamma$, then $A+B-B$ contains $\\{\gamma\in\Gamma:\mu((A\cdot
D)\cap T^{\gamma}D)>0\\}$.

###### Proof.

Note that $\mu(T^{a}D\cap T^{\gamma}D)>0$ iff $\mu(D\cap T^{\gamma-a}D)>0$,
which implies $\gamma-a\in B-B$, meaning $\gamma\in a+B-B$. It follows that
$\mu((A\cdot D)\cap T^{\gamma}D)>0$ implies $\gamma\in A+B-B$. ∎

###  5.4. Proof of Proposition 4.1

From now on we fix:

  1. $\bullet$

$A$, $B\subseteq\Gamma$ having $d^{*}(A)>0$, $d^{*}(B)>0$,

  2. $\bullet$

a Følner sequence $\mathbf{\Phi}$ satisfying $d_{\mathbf{\Phi}}(A)=d^{*}(A)$,

  3. $\bullet$

an ergodic $\Gamma$-system $\mathbf{X}=(X,\mathscr{X},\mu,T)$ and a set
$D\subseteq X$ satisfying the conclusion of Lemma 5.2,

  4. $\bullet$

the Kronecker factor $\mathbf{Z}=(Z,\mathcal{Z},m,R_{\rho})$ of
$(X,\mathscr{X},\mu,T)$, with factor map $\pi:X\to Z$. Note that
$\rho:\Gamma\to Z$ is a homomorphism with $\rho(\Gamma)$ dense in $Z$.

  5. $\bullet$

a function $g:Z\to[0,1]$ satisfying $g\circ\pi=P_{\mathcal{Z}}1_{D}$
$\mu$-almost everywhere.

Here $P_{\mathcal{Z}}:L^{2}(\mu)\to L^{2}(\mu)$ is the orthogonal projection
onto the space of $\pi^{-1}(\mathcal{Z})$-measurable functions. As a special
case of Equation (5.4), we get

(5.6) $\displaystyle
g(\pi(x)+\rho(\gamma))=P_{\mathcal{Z}}1_{D}(T^{\gamma}x)\text{ for $\mu$-a.e.
}x\text{ and all }\gamma\in\Gamma.$

Furthermore, if $F$ is $\pi^{-1}(\mathcal{Z})$-measurable, meaning
$F=\tilde{F}\circ\pi$ for some $\tilde{F}\in L^{2}(m)$, then Equation (5.5)
implies

(5.7) $\int F\cdot 1_{D}\circ T^{-\gamma}\,d\mu=\int\tilde{F}\cdot g\circ
R_{\rho}^{-\gamma}\,dm=\int\tilde{F}(z)\cdot g(z-\rho(\gamma))\,dm(z)$

for all $\gamma\in\Gamma$.

The lemmas and proofs in the remainder of this section will refer to the
objects defined above. Our goal, in light of Lemma 5.2, is to describe those
$\gamma$ for which $\mu((A\cdot D)\cap T^{-\gamma}D)>0$. Our approach is
similar to the proofs of Proposition 3.2 of [4] and Proposition 4.2 of [5].

In order to understand $A\cdot D$, we consider averages of functions supported
on $A\cdot D$. We have fixed a Følner sequence
$\mathbf{\Phi}=(\Phi_{n})_{n\in\mathbb{N}}$ such that
$d_{\mathbf{\Phi}}(A)=d^{*}(A)$. Consider the sets $A_{n}:=A\cap\Phi_{n}$.
Observe that $1_{D}\circ T^{-a}$ is supported on $A\cdot D$ for each $a\in A$,
so each average

(5.8) $F_{n}:=\frac{1}{|\Phi_{n}|}\sum_{a\in A_{n}}1_{D}\circ T^{-a}$

is supported on $A\cdot D$, and every weak $L^{2}(\mu)$ limit of these
averages is also supported on $A\cdot D$. Passing to a subsequence of
$\mathbf{\Phi}$, we may assume that the limit of $(F_{n})_{n\in\mathbb{N}}$
exists. Analyzing this limit will lead to the following lemma. See Section 3.2
for the definitions of $f*g$ and $g_{-}.$

######  Lemma 5.3.

  1. (i)

The set $A\cdot D$ supports a function of the form $f*g\circ\pi$, where
$f:Z\to[0,1]$ has $\int f\,dm=d^{*}(A)$, $f$ is supported on
$\overline{\rho(A)}$, and $g$ is defined above.

  2. (ii)

$\\{\gamma\in\Gamma:\mu((A\cdot D)\cap
T^{\gamma}D)>0\\}\supseteq\\{\gamma:f*g*g_{-}(\rho(\gamma))>0\\}$.

###### Proof.

To prove (i), let $F$ be a weak $L^{2}(\mu)$ limit of the sequence
$(F_{n})_{n\in\mathbb{N}}$, defined in Equation (5.8). Then $F$ is supported
on $A\cdot D$, and by Lemma 5.4, $F$ has the form $f*g\circ\pi$, where $f$ is
as described in Part (ii) of that lemma.

Proof of Part (ii). We must prove the implication

(5.9) $f*g*g_{-}(\rho(\gamma))>0\implies\mu((A\cdot D)\cap T^{\gamma}D)>0.$

Suppose $\gamma\in\Gamma$ satisfies

(5.10) $\int(f*g\circ\pi)\cdot 1_{D}\circ T^{-\gamma}\,d\mu>0.$

Equation (5.7) implies the integral in Inequality (5.10) is positive if and
only if $\int f*g(z)\cdot g(z-\rho(\gamma))\,dm(z)>0,$ meaning
$f*g*g_{-}(\rho(\gamma))>0$. By Part (i), inequality (5.10) implies
$\mu((A\cdot D)\cap T^{\gamma}D)>0$, so we have proven the implication (5.9).
∎

###### Proof of Proposition 4.1.

Proposition 4.1 now follows from Lemmas 5.2 and 5.3. ∎

######  Lemma 5.4.

If the weak $L^{2}(\mu)$ limit $F$ of the sequence

$F_{n}:=\frac{1}{|\Phi_{n}|}\sum_{a\in A_{n}}1_{D}\circ T^{-a}$

exists and $g:Z\to[0,1]$ is the function satisfying Equation (5.6), then

  1. (i)

$F$ is $\pi^{-1}(\mathcal{Z})$-measurable, and

$\displaystyle F=\lim_{n\to\infty}\frac{1}{|\Phi_{n}|}\sum_{a\in A_{n}}g\circ
R_{\rho}^{-1}\circ\pi\text{ weakly in }L^{2}(\mu).$

  2. (ii)

$F=(f*g)\circ\pi$, where $f:Z\to[0,1]$ satisfies $\int f\,dm=d^{*}(A)$ and $f$
is supported on $\overline{\rho(A)}$.

###### Proof.

Part (i) is a consequence of the proof of Corollary 2.7 of [5]. To prove Part
(ii), it suffices to show that

(5.11) $\int F\cdot\psi\,d\mu=\int(f*g)\circ\pi\cdot\psi\,d\mu\text{\ \ for
every }\psi\in L^{2}(\mu).$

Part (i) already proves Equation (5.11) for those $\psi$ orthogonal to the
$\pi^{-1}(\mathcal{Z})$-measurable functions, so we may assume that $\psi$ is
$\pi^{-1}(\mathcal{Z})$-measurable, meaning $\psi=\tilde{\psi}\circ\pi$ for
some $\tilde{\psi}\in L^{2}(m)$. It suffices to establish the identity (5.11)
for an $L^{2}(\mu)$-dense set of functions $\psi$, so we may assume that
$\psi=\tilde{\psi}\circ\pi$, where $\tilde{\psi}:Z\to\mathbb{C}$ is
continuous. Let $F$ be a weak $L^{2}(\mu)$ limit of the
$(F_{n})_{n\in\mathbb{N}}$, and let $\nu$ be a measure as in the conclusion of
Lemma 5.5, so that $d\nu=f\,dm$ for some $f$ as in the conclusion of the
lemma. Then we apply Part (i) to compute $F$:

(5.12) $\displaystyle\int F\cdot\psi\,d\mu$
$\displaystyle=\lim_{n\to\infty}\frac{1}{|\Phi_{n}|}\sum_{a\in A_{n}}\int
g(\pi(x)-\rho(a))\cdot\tilde{\psi}(\pi(x))\,d\mu(x)$ (5.13)
$\displaystyle=\lim_{n\to\infty}\frac{1}{|\Phi_{n}|}\sum_{a\in A_{n}}\int
g(\pi(x))\cdot\tilde{\psi}(\pi(x)+\rho(a))\,d\mu(x)$ (5.14)
$\displaystyle=\lim_{n\to\infty}\int g(z)\cdot\frac{1}{|\Phi_{n}|}\sum_{a\in
A_{n}}\tilde{\psi}(z+\rho(a))\,dm(z).$

We used Equations (5.3) and (5.7) to get the second and third lines. Applying
Lemma 5.5 to evaluate the limit in Equation (5.14), we find

$\lim_{n\to\infty}\frac{1}{|\Phi_{n}|}\sum_{a\in
A_{n}}\tilde{\psi}(z+\rho(a))=\int\tilde{\psi}(z+w)f(w)\,dm(w)\ \text{ for all
}z\in Z,$

where $f$ is as described above. We then evaluate, continuing from Equation
(5.14),

$\displaystyle\int F\cdot\psi\,d\mu$ $\displaystyle=\int
g(z)\cdot\int\tilde{\psi}(z+w)f(w)\,dm(w)\,dm(z)$ $\displaystyle=\int\int
f(w)g(z-w)\,dm(w)\,\tilde{\psi}(z)\,dm(z)$ $\displaystyle=\int
f*g(z)\cdot\tilde{\psi}(z)\,dm(z)$
$\displaystyle=\int(f*g)\circ\pi\cdot\psi\,d\mu,$

establishing Equation (5.11). The second line uses Fubini and translation
invariance of $dm$, the third is just the definition of $f*g$, and the last
line is the special case of Equation (5.5) with $\gamma=0$. ∎

######  Lemma 5.5.

Consider the Borel measures $\nu_{n}$ on $Z$ given by

$\displaystyle\int\psi\,d\nu_{n}=\frac{1}{|\Phi_{n}|}\sum_{\gamma\in
A_{n}}\psi(\rho(\gamma))$ $\displaystyle\text{ for }\psi:Z\to\mathbb{C}.$

Let $\nu$ be a $\text{weak}^{*}$ limit of the $\nu_{n}$. Then $\nu$ is
absolutely continuous with respect to Haar measure $m$, and

  1. (i)

$d\nu=f\,dm$, where $f:Z\to[0,1]$,

  2. (ii)

$\nu(Z)=d^{*}(A)$,

  3. (iii)

$f$ is supported on $\overline{\rho(A)}$.

###### Proof.

Parts (i) and (ii) are established in the proof of Lemma 2.11 of [5]. Part
(iii) follows from the fact that $\int\psi\,d\nu=0$ whenever
$\psi:Z\to\mathbb{C}$ is a continuous function which vanishes on
$\overline{\rho(A)}$. ∎

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