# Kolmogorov widths of Besov classes $B^{1}_{1,\theta}$ and products of
octahedra

Yuri Malykhin

###### Abstract

In this paper we find the orders of decay for Kolmogorov widths of some Besov
classes related to $W^{1}_{1}$ (the behaviour of the widths for $W^{1}_{1}$
remains unknown):

$d_{n}(B^{1}_{1,\theta}[0,1],L_{q}[0,1])\asymp
n^{-1/2}\log^{\max(\frac{1}{2},1-\frac{1}{\theta})}n,\quad 2<q<\infty.$

The proof relies on the lower bound for widths of product of octahedra in a
special norm (maximum of two weighted $\ell_{q}$ norms). This bound
generalizes the theorem of B.S. Kashin on widths of octahedra in
$\ell_{q}^{N}$.

##  1 Introduction

Let us recall the definition of widths. Let $X$ be a normed space and
$K\subset X$. The Kolmogorov $n$-width of $K$ in $X$ is the following quantity

$d_{n}(K,X):=\inf_{\begin{subarray}{c}Q_{n}\subset X\\\ \dim Q_{n}\leqslant
n\end{subarray}}\sup_{x\in K}\inf_{y\in Q}\|x-y\|_{X},$

where $\inf_{Q_{n}}$ is taken over all linear subspaces in $X$ having
dimension at most $n$.

The main problem of the widths theory is to find the order of decay of
$d_{n}(W,X)$ at $n\to\infty$ for functional classes and finite-dimensional
sets. One of interesting cases is the family of Sobolev spaces
$\mathcal{W}^{r}_{p}$ consisting of functions $f$ with a finite norm
$\|f\|_{p}+\|f^{(r)}\|_{p}$ and Sobolev classes $W^{r}_{p}$ (unit balls in
corresponding spaces). The order of $d_{n}(W^{r}_{p}[0,1],L_{q}[0,1])$ is
known for all $r\in\mathbb{N}$ and $1\leqslant p,q\leqslant\infty$ (V.M.
Tikhomirov, B.S. Kashin, E.D. Gluskin, R.S. Ismagilov — see [10], Chapters 13,
14, and the survey [8], §4.3), except the case $r=1$, $p=1$, $q\in(2,\infty)$.
There is a logarithmic gap [9, 18]:

$c(q)n^{-1/2}\log^{1/2}n\leqslant d_{n}(W^{1}_{1}[0,1],L_{q}[0,1])\leqslant
C(q)n^{-1/2}\log n.$

The order of approximation is known only for special approximation subspaces
$Q_{n}=\mathrm{span}\\{\exp(im_{j}x)\\}_{j=1}^{n}$, see. [12]:
$d_{n}^{T}(W^{1}_{1}[0,1],L_{q}[0,1])\asymp n^{-1/2}\log n$.

Sobolev spaces are related to Besov spaces. Let us give the definition,
following [6], Chapter 2, §10. Given parameters $r>0$, $p,\theta\in[1,\infty]$
and a (finite or infinite) segment $I\subset\mathbb{R}$, we define the Besov
space $\mathcal{B}^{r}_{p,\theta}(I)$ as the space of functions $f\in
L_{p}(I)$ having a finite semi-norm (with $s=1+\lfloor r\rfloor$)

$|f|_{B^{r}_{p,\theta}(I)}:=\begin{cases}\int_{0}^{\infty}(t^{-r}\omega_{s}(f,t)_{p}^{\theta})^{1/\theta}\frac{dt}{t},&\quad
1\leqslant\theta<\infty,\\\
\sup_{t>0}t^{-r}\omega_{s}(f,t)_{p},&\quad\theta=\infty.\end{cases}$

Here $\omega_{s}(f,t)_{p}:=\sup_{0<h\leqslant
t}\|\Delta_{h}^{s}(f,\cdot)\|_{L_{p}(I\cap(I-sh))}$ is the usual modulus of
smoothness. The norm in the Besov space is defined as
$\|f\|_{L_{p}(I)}+|f|_{B^{r}_{p,\theta}(I)}$. The Besov class
$B^{r}_{p,\theta}(I)$ is the unit ball in the corresponding space.

######  Remark.

One can define Besov space on domains $I\neq\mathbb{R}$ as the restriction of
functions $f\in B^{r}_{p,\theta}(\mathbb{R})$ to $I$. Then the norm of $g$ is
defined as $\inf\\{\|f\|\colon\left.f\right|_{I}=g\\}$. For segments
$I\subset\mathbb{R}$ these definitions (and corresponding norms) are
equivalent, see, e.g., [4], §1.11, Theorem 1.118.

There are embeddings
$\mathcal{B}^{r}_{p,\min(p,2)}\hookrightarrow\mathcal{W}^{r}_{p}\hookrightarrow\mathcal{B}^{r}_{p,\max(p,2)}$
for $1<p<\infty$ and
$\mathcal{B}^{r}_{p,1}\hookrightarrow\mathcal{W}^{r}_{p}\hookrightarrow\mathcal{B}^{r}_{p,\infty}$
for $p=1$, $p=\infty$. As a corollary, we have $c_{1}B^{1}_{1,1}[0,1]\subset
W^{1}_{1}[0,1]\subset c_{2}B^{1}_{1,\infty}[0,1]$. This gives us motivation to
study widths of $B^{1}_{1,\theta}$.

Of course, widths of Besov classes are interesting by themselves. Let us
mention the papers of Ding Dung [14], A.S. Romanyuk [15], E.M. Galeev [16];
see also the survey [8], §4.3. The case $B^{1}_{1,\theta}$ in $L_{q}$, $q>2$,
as far as author knows, was not studied before.

######  Theorem 1.

For any $q\in(2,\infty)$ there exist $c_{q}>0$ and $C_{q}>0$, such that for
all $1\leqslant\theta\leqslant\infty$ and $n=2,3,\ldots$ the inequalities
hold:

$c_{q}n^{-1/2}\log^{1/2}n\leqslant
d_{n}(B^{1}_{1,\theta}[0,1],L_{q}[0,1])\leqslant
C_{q}n^{-1/2}\log^{1/2}n,\quad 1\leqslant\theta\leqslant 2,$
$c_{q}n^{-1/2}\log^{1-1/\theta}n\leqslant
d_{n}(B^{1}_{1,\theta}[0,1],L_{q}[0,1])\leqslant
C_{q}n^{-1/2}\log^{1-1/\theta}n,\quad 2\leqslant\theta\leqslant\infty.$

Unfortunately, this gives us no new information about $W^{1}_{1}$.

The proof is based on the following finite-dimensional result. Consider a
weighted norm in $\mathbb{R}^{N}$:

$\|x\|_{\ell_{q,w}^{N}}:=\left\\{\sum_{i=1}^{N}w_{i}|x_{i}|^{q}\right\\}^{1/q}.$

Suppose that the set of coordinates is divided into $m$ blocks:
$\\{1,\ldots,N\\}=\sqcup_{s=1}^{m}\Delta_{s}$, $|\Delta_{s}|=N_{s}$. Let us
take the octahedron $B_{1}^{N_{s}}$ in each block, and consider their
Cartesian product

$\prod_{s=1}^{m}B_{1}^{N_{s}}=\\{x\in\mathbb{R}^{N}\colon\sum_{i\in\Delta_{s}}|x_{i}|\leqslant
1,\;s=1,\ldots,m\\}.$

######  Theorem 2.

Let there are weights $w_{1},\ldots,w_{N}\geqslant 0$, division into $m$
blocks $\\{1,\ldots,N\\}=\sqcup_{s=1}^{m}\Delta_{s}$, $N_{s}=|\Delta_{s}|$,
and a number $n\in\mathbb{N}$, such that the following conditions hold:

  * 1)

the weights are sufficiently small: $\max\limits_{1\leqslant i\leqslant
N}w_{i}\leqslant(4n)^{-1}\sum_{i=1}^{N}w_{i}$;

  * 2)

the weighted ratio of each block is not small:

$\nu_{s}:=\frac{\sum_{i\in\Delta_{s}}w_{i}}{\sum_{i=1}^{N}w_{i}}\geqslant\frac{C\log
2m}{n},\quad s=1,\ldots,m.$

where $C$ is enough large absolute constant.

Suppose that for $q\in[2,\infty)$ and $h>0$ the norm in the space
$X=(\mathbb{R}^{N},\|\cdot\|)$ satisfies

$\|x\|_{X}\geqslant\max(\|x\|_{\ell_{{q},w}^{N}},h\|x\|_{\infty}),\quad
x\in\mathbb{R}^{N}.$

Then the Kolmogorov width of the product of octahedra in $X$ satisfies the
inequality

$d_{n}(\prod_{s=1}^{m}B_{1}^{N_{s}},X)\geqslant\min(c_{q}\,(n\max_{1\leqslant
s\leqslant m}\nu_{s})^{-1/2}(\sum_{i=1}^{N}w_{i})^{1/q},h/2).$

We note that the condition 2) implies that there are not so many blocks:
$Cm\log(2m)\leqslant n$.

Let us mention some corollaries of the Theorem 2. Here and after by
intersection $X\cap Y$ of two normed spaces we mean the space with the norm
$\|x\|_{X\cap Y}:=\max(\|x\|_{X},\|x\|_{Y})$. If we take $q_{2}\geqslant 1$,
then
$\|x\|_{\ell_{{q_{2}},w}^{N}}\geqslant\min_{i}w_{i}^{1/q_{2}}\|x\|_{\infty}$.
Suppose that the sum of weights in each block is constant (hence
$\nu_{s}=m^{-1}$), then the bound in the theorem becomes (where we put
$q_{1}:=q$ for more symmetry):

$d_{n}(\prod_{s=1}^{m}B_{1}^{N_{s}},\ell_{{q_{1}},w}^{N}\cap\ell_{{q_{2}},w}^{N})\geqslant\min(c_{q_{1}}\sqrt{\frac{m}{n}}(\sum_{i=1}^{N}w_{i})^{1/q_{1}},\frac{1}{2}\min
w_{i}^{1/q_{2}}).$ (1)

We will use the bound in the form (1) to prove Theorem 1; we shall take
$q_{1}=2$ and $q_{2}$ equal to $q$ from Theorem 1.

If $q_{1}=q_{2}=q\in[2,\infty)$ in (1) then we get lower bound for the width
of product of octahedra in $\ell_{q,w}^{N}$. Moreover, if all blocks are equal
and $w_{i}=1$, then we get

$d_{n}(\prod_{s=1}^{m}B_{1}^{N/m},\ell_{q}^{N})\geqslant\min(c_{q}\sqrt{\frac{m}{n}}N^{1/q},\frac{1}{2}),\quad\mbox{if
$Cm\log(2m)\leqslant n\leqslant N/4$}.$

For $m=1$ this was proven in the paper of B.S. Kashin [1], there was also
noted that the bound is sharp:

$\frac{1}{4}\min(\frac{N^{1/q}}{\sqrt{n}},1)\leqslant
d_{n}(B_{1}^{N},\ell_{q}^{N})\leqslant\min(c_{q}\frac{N^{1/q}}{\sqrt{n}},1),\quad
n<\frac{N}{2},\;q\in(2,\infty).$ (2)

Our proof of 2 is the development of a method from [1].

The product of octahedra had already appeared in the context of widths of
functional classes, see, e.g. [13], [19].

##  2 Proofs

We will prove Theorem 2 and then derive Theorem 1.

###### Proof.

W.l.o.g. $\|x\|_{X}=\max(\|x\|_{\ell_{{q},w}^{N}},h\|x\|_{\infty})$. We denote
$B:=\prod_{s=1}^{m}B_{1}^{N_{s}}$, $d_{n}:=d_{n}(B,X)$, and let $L_{n}$ be the
extremal subspace for that width. If $d_{n}\geqslant h/2$, the proof is done,
so we will assume that $d_{n}<h/2$.

We are interested in the approximation of the extreme points of $B$; it is
easy to see that $\operatorname{extr}B=\\{\sum_{s=1}^{m}\pm e_{i_{s}}\colon
i_{s}\in\Delta_{s}\\}$. Let us show that for any $u\in\operatorname{extr}B$
there is $v=v(u)\in L_{n}$ such that

$\|u-v\|_{X}\leqslant 2d_{n},$ (3)
$u_{i}=v_{i}\quad\forall\,i\in\operatorname{supp}u.$ (4)

Fix $u\in\operatorname{extr}B$ and take $v_{0}\in L_{n}$, such that
$\|u-v_{0}\|_{X}\leqslant d_{n}$. Write the error $u-v_{0}$ as
$u_{1}+\beta_{1}$, where
$\operatorname{supp}u_{1}\subset\operatorname{supp}u$,
$\operatorname{supp}\beta_{1}\cap\operatorname{supp}u=\varnothing$. We have
$\|\beta_{1}\|\leqslant d_{n}$,

$\|u_{1}\|_{\infty}\leqslant h^{-1}\|u_{1}\|_{X}\leqslant d_{n}/h\leqslant
1/2,$

and the condition on $\operatorname{supp}u_{1}$ implies that
$u_{1}\in\frac{1}{2}B$. Hence, the vector $u_{1}$ may be approximated by
$v_{1}\in L_{n}$ with an error $\leqslant\frac{1}{2}d_{n}$. Further, we write
$u_{1}-v_{1}=u_{2}+\beta_{2}$,
$\operatorname{supp}u_{2}\subset\operatorname{supp}u$ and approximate $u_{2}$.
If we repeat this process, we will get the approximation of $u$ by the sum
$v_{0}+v_{1}+\ldots\in L_{n}$ with the error
$\|\beta_{1}+\beta_{2}+\ldots\|\leqslant 2d_{n}$.

Let us fix a mapping $v(\cdot)\colon\operatorname{extr}B\to L_{n}$, which
satisfies (3) and (4).

We can normalize the weights by $\sum_{i=1}^{N}w_{i}=1$. Let $p_{i}:=2nw_{i}$.
Let $\mathsf{\Omega}\subset\\{1,\ldots,N\\}$ be a random subset, such that any
point $i$ gets in $\mathsf{\Omega}$ independently of others with probability
$p_{i}$. Condition 1) of the Theorem gives us $p_{i}\in[0,1]$ (even
$p_{i}\leqslant 1/2$; we will use that later). The choice of the probabilites
proportional to the weights allows us to write weighted the norm as the
expectation:

$\mathsf{E}\sum_{i\in\mathsf{\Omega}}|x_{i}|^{q}=2n\|x\|_{\ell_{{q},w}^{N}}^{q}.$

Also, $\mathsf{E}|\mathsf{\Omega}|=\sum p_{i}=2n$ and
$\mathsf{E}|\mathsf{\Omega}\cap\Delta_{s}|=2n\nu_{s}$.

Let $\mathsf{u}$ be a random point in $\operatorname{extr}B$ (equiprobable)
independent of $\mathsf{\Omega}$. In other words, in each block $\Delta_{s}$
one random coordinate of $\mathsf{u}$ equals $\pm 1$ and others are zero. Put
$\mathsf{v}=v(\mathsf{u})$. Consider the random variable

$\xi:=\sum_{i\in\mathsf{\Omega}}|\mathsf{u}_{i}-\mathsf{v}_{i}|^{q}\cdot\mathbf{1}\\{\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega}\\}$

and the conditional expectation

$K:=\mathsf{E}(\sum_{i\in\mathsf{\Omega}}|\mathsf{u}_{i}-\mathsf{v}_{i}|^{q}\,|\,\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega})=\frac{\mathsf{E}\xi}{\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega})}.$

The meaning of the condition
$\\{\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega}\\}$ is that it allows
us to come to a low-dimensional case:
$|\mathsf{\Omega}|\approx\mathsf{E}|\mathsf{\Omega}|=2n$.

We will bound $K$ in two ways. Fix $u^{\circ}$ and average over
$\mathsf{\Omega}$:

$\mathsf{E}(\xi|\mathsf{u}=u^{\circ})=\sum_{i=1}^{N}|u^{\circ}_{i}-v^{\circ}_{i}|^{q}\cdot\mathsf{P}(\operatorname{supp}u^{\circ}\subset\mathsf{\Omega},\;i\in\mathsf{\Omega}).$

If $i\in\operatorname{supp}u^{\circ}$, then by (4) the corresponding term is
zero. Otherwise, by the independence,

$\frac{\mathsf{P}(\operatorname{supp}u^{\circ}\subset\mathsf{\Omega},\;i\in\mathsf{\Omega})}{\mathsf{P}(\operatorname{supp}u^{\circ}\subset\mathsf{\Omega})}=P(i\in\mathsf{\Omega})=2nw_{i}.$

We sum over $i$ and obtain

$\frac{\mathsf{E}(\xi|\mathsf{u}=u^{\circ})}{\mathsf{P}(\operatorname{supp}u^{\circ}\subset\mathsf{\Omega})}=2n\sum_{i=1}^{N}w_{i}|u^{\circ}_{i}-v^{\circ}_{i}|^{q}=2n\|u^{\circ}-v^{\circ}\|_{\ell_{{q},w}^{N}}^{q}\ll
nd_{n}^{q}.$

Then we multiply by the probability, average over $u^{\circ}$ and obtain

$\mathsf{E}\xi\ll
nd_{n}^{q}\cdot\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega}),\quad
K\ll nd_{n}^{q}.$

Now we fix $\Omega^{\circ}$ and average over $u$:

$\mathsf{E}(\xi|\mathsf{\Omega}=\Omega^{\circ})=\mathsf{E}\left(\left.\sum_{i\in\Omega^{\circ}}|\mathsf{u}_{i}-\mathsf{v}_{i}|^{q}\;\right|\operatorname{supp}\mathsf{u}\subset\Omega^{\circ}\right)\cdot\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\Omega^{\circ}).$

Let us call a set $\Omega$ regular if the cardinalities of the sets
$\Omega_{s}:=\Omega\cap\Delta_{s}$ satisfy
$\frac{3}{2}n\nu_{s}\leqslant|\Omega_{s}|\leqslant An\nu_{s}$ for all $s$; the
constant $A$ is large enough (see later).

Suppose that $\Omega^{\circ}$ is regular. Obviously,
$\frac{3}{2}n\leqslant|\Omega^{\circ}|\leqslant An$. By
$\widetilde{\mathsf{u}}$ we denote the random vector with the conditional
distribution
$\mathsf{Law}(\widetilde{\mathsf{u}})=\mathsf{Law}(\mathsf{u}|\operatorname{supp}\mathsf{u}\subset\Omega^{\circ})$.
Let $\widetilde{\mathsf{v}}=v(\widetilde{\mathsf{u}})$. We have

$\mathsf{E}\left(\left.\sum_{i\in\Omega^{\circ}}|\mathsf{u}_{i}-\mathsf{v}_{i}|^{q}\;\right|\operatorname{supp}\mathsf{u}\subset\Omega^{\circ}\right)=\mathsf{E}\|\widetilde{\mathsf{u}}-\widetilde{\mathsf{v}}\|_{\ell_{q}(\Omega^{\circ})}^{q}.$

We can consider $\widetilde{\mathsf{u}}$ as a random vector in
$\mathbb{R}^{\Omega^{\circ}}$. It is easy to see that $\widetilde{\mathsf{u}}$
takes values (equiprobable) in the set
$\operatorname{extr}\prod_{s=1}^{m}B_{1}^{|\Omega_{s}^{\circ}|}$; in other
words, in each set $\Omega^{\circ}_{s}$ one random coordinate is $\pm 1$ and
others are zero. We can pass from $q$-mean to $2$-mean and from
$\ell_{q}(\Omega^{\circ})$ metric to $\ell_{2}(\Omega^{\circ})$ (here we use
that $q\geqslant 2$):

$(\mathsf{E}\|\widetilde{\mathsf{u}}-\widetilde{\mathsf{v}}\|_{q}^{q})^{1/{q}}\geqslant(\mathsf{E}\|\widetilde{\mathsf{u}}-\widetilde{\mathsf{v}}\|_{q}^{2})^{1/{2}}\gg
n^{1/{q}-1/2}(\mathsf{E}\|\widetilde{\mathsf{u}}-\widetilde{\mathsf{v}}\|_{2}^{2})^{1/2}.$

The estimate for the approximation in Euclid metric is standard (a variation
of Eckart-Young theorem).

######  Lemma 1 (See, e.g., [11]).

Let $\mathsf{z}$ be a random vector in Hilbert space $H$ with
$\mathsf{E}\|z\|_{H}^{2}<\infty$ and in some orthonormal basis
$\\{\varphi_{i}\\}$ we have
$\mathsf{E}\langle\mathsf{z},\varphi_{i}\rangle\langle\mathsf{z},\varphi_{j}\rangle=0$
for $i\neq j$. Let
$\sigma_{i}:=(\mathsf{E}\langle\mathsf{z},\varphi_{i}\rangle^{2})^{1/2}$ and
$\sigma_{1}^{*}\geqslant\sigma_{2}^{*}\geqslant\ldots$ be the non-increasing
permutation of $(\sigma_{i})$. Then for any $n$-dimensional subspace
$L_{n}\subset H$ the inequality holds

$\mathsf{E}\rho(\mathsf{z},L_{n})_{H}^{2}\geqslant\sum_{k>n}(\sigma_{k}^{*})^{2},\quad\mbox{where
$\rho(x,L_{n})_{H}:=\inf_{y\in L}\|x-y\|_{H}$}.$

In our case one can take the standard basis $\\{e_{i}\\}$, because
$\mathsf{E}\widetilde{\mathsf{u}}_{i}\widetilde{\mathsf{u}}_{j}=0$;
$\sigma_{i}^{2}=\mathsf{E}\widetilde{\mathsf{u}}_{i}^{2}=|\Omega_{s}^{\circ}|^{-1}\asymp(n\nu_{s})^{-1}$,
$i\in\Delta_{s}$. Denote $\nu:=\max\nu_{s}$, then

$\mathsf{E}\|\widetilde{\mathsf{u}}-\widetilde{\mathsf{v}}\|_{2}^{2}\geqslant\mathsf{E}\rho(\widetilde{\mathsf{u}},L_{n})_{2}^{2}\geqslant\sum_{k=n+1}^{3n/2}(\sigma_{k}^{*})^{2}\gg\nu^{-1}.$

Therefore, for regular $\Omega^{\circ}$ we have

$\mathsf{E}(\xi|\mathsf{\Omega}=\Omega^{\circ})\gg\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\Omega^{\circ})\cdot
n^{1-q/2}\cdot\nu^{-q/2}.$

Averaging over $\Omega^{\circ}$ we obtain

$\mathsf{E}\xi\gg
n^{1-q/2}\nu^{-q/2}\cdot\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega},\;\mathsf{\Omega}\mbox{
is regular}).$

Later we will show that one can separate conditions in $\mathsf{P}(\cdot)$:

$\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega},\;\mathsf{\Omega}\mbox{
is
regular})\geqslant\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega})\mathsf{P}(\mathsf{\Omega}\mbox{
is regular}).$ (5)

It is easy to check that the probability that $\mathsf{\Omega}$ is regular is
$\geqslant 1/2$. To do this we apply Bernstein’s inequality:

$\mathsf{P}(|\sum_{i=1}^{k}X_{i}|>t)\leqslant
2\exp(-\frac{t^{2}}{2(\sigma^{2}+Mt/3)}),$

where $X_{1},\ldots,X_{k}$ are independent r.v. with zero average,
$M:=\max_{i}\|X_{i}\|_{\infty}$, $\sigma^{2}:=\sum_{i}\mathsf{E}X_{i}^{2}$.
Fix $s$, put $X_{i}=\mathbf{1}\\{i\in\mathsf{\Omega}_{s}\\}-p_{i}$,
$i\in\Delta_{s}$ (recall that
$\mathsf{\Omega}_{s}:=\mathsf{\Omega}\cap\Delta_{s}$), use that $M\leqslant
1$, $\sigma^{2}\leqslant\sum_{i\in\Delta_{s}}p_{i}=2n\nu_{s}$ to obtain

$\mathsf{P}(||\mathsf{\Omega}_{s}|-2n\nu_{s}|>n\nu_{s}/2)\leqslant
2\exp(-\frac{(n\nu_{s}/2)^{2}}{2(2n\nu_{s}+n\nu_{s}/6)})\leqslant
2\exp(-cn\nu_{s}).$

The condition 2) of the Theorem implies that this probability is less than
$\frac{1}{2m}$. Hence, the probability of irregularity of $\mathsf{\Omega}$ is
less than $1/2$. So,

$\mathsf{E}\xi\gg
n^{1-q/2}\nu^{-q/2}\mathsf{P}(\operatorname{supp}\mathsf{u}\subset\mathsf{\Omega}),\quad
K\gg n^{1-q/2}\nu^{-q/2}.$

Finally, we compare upper and lower bounds on $K$ and obtain the required
bound for $d_{n}$.

It remains to prove (5). We will use one technical lemma. Here we write
$\\#\Omega$ instead of $|\Omega|$ for convenience.

######  Lemma 2.

Let $\delta\in(0,1)$, $\mathsf{\Omega}$ is a random subset of
$\\{1,\ldots,N\\}$, and each point $i$ gets in $\mathsf{\Omega}$ independently
of others with probability $p_{i}\leqslant 1/2$. If
$\omega:=\mathsf{E}\\#\mathsf{\Omega}=\sum_{i=1}^{N}p_{i}$ is large enough
($\omega\geqslant c(\delta)$), then

$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{R})\geqslant\mathsf{E}\\#\mathsf{\Omega},\quad\mbox{where
$\mathcal{R}:=\\{(1-\delta)\omega\leqslant\\#\mathsf{\Omega}\leqslant
5\omega\\}$}.$

Let us derive (5) from lemma. It is enough to prove the inequality for each
block (because of the independence):

$\mathsf{P}(\operatorname{supp}\mathsf{u}\cap\Delta_{s}\subset\mathsf{\Omega}\cap\Delta_{s}\big{|}3n\nu_{s}/2\leqslant|\mathsf{\Omega}_{s}|\leqslant
5n\nu_{s})\geqslant\mathsf{P}(\operatorname{supp}\mathsf{u}\cap\Delta_{s}\subset\mathsf{\Omega}\cap\Delta_{s}).$

The right side of the inequality is
$N_{s}^{-1}\mathsf{E}|\mathsf{\Omega}\cap\Delta_{s}|$, and left side is equal
to the corresponding conditional expectation. It remains to apply the Lemma
for $\mathsf{\Omega}\cap\Delta_{s}$.

Let us prove this lemma. The probability space is a union of non-intersecting
events $\mathcal{R}$,
$\mathcal{C}_{-}:=\\{\\#\mathsf{\Omega}|<(1-\delta)\omega\\}$ and
$\mathcal{C}_{+}:=\\{\\#\mathsf{\Omega}>5\omega\\}$. Denote
$\mathcal{C}=\mathcal{C}_{-}\cup\mathcal{C}_{+}$. We have to prove that
$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{R})\geqslant\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C})$.
Using Bernstein’s inequality we get

$\mathsf{P}(\\#\mathsf{\Omega}\in[(1-\delta/3)\omega,(1+\delta/3)\omega])\geqslant
1-2\exp(-c_{\delta}\omega)>9/10$

for large $\omega$. This is also true conditioning on $\mathcal{R}$, so

$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{R})\geqslant\frac{1}{10}(1-\delta)\omega+\frac{9}{10}(1-\delta/3)\omega>(1-\delta/2)\omega.$

Let us bound
$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C})=\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C}_{-})\mathsf{P}(\mathcal{C}_{-}|\mathcal{C})+\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C}_{+})\mathsf{P}(\mathcal{C}_{+}|\mathcal{C})$.

The first term is not more than
$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C}_{-})\leqslant(1-\delta)\omega$. The
second term is equal to

$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C}_{+})\mathsf{P}(\mathcal{C}_{+}|\mathcal{C})=\frac{\mathsf{E}\\#\mathsf{\Omega}\mathbf{1}_{\mathcal{C}_{+}}}{\mathsf{P}(\mathcal{C})}=\frac{1}{\mathsf{P}(\mathcal{C})}\sum_{k>5\omega}k\mathsf{P}(\\#\mathsf{\Omega}=k).$

Note that
$\mathsf{P}(\mathcal{C})\geqslant\mathsf{P}(\mathsf{\Omega}=\varnothing)=\prod(1-p_{i})\geqslant\prod
4^{-p_{i}}=4^{-\omega}$ (we use that $p_{i}\leqslant 1/2$), hence the second
term is less or equal than

$4^{\omega}\sum_{k>5\omega}k\mathsf{P}(\\#\mathsf{\Omega}=k)\leqslant
4^{\omega}\sum_{k>5\omega}k\exp(-ck)\leqslant C\omega
e^{-\omega}<\frac{\delta}{2}\omega$

for large $\omega$. Summing up, we obtain
$\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{C})\leqslant(1-\delta/2)\omega<\mathsf{E}(\\#\mathsf{\Omega}|\mathcal{R})$.
The lemma and the Theorem are proven. ∎

Before we proceed to the proof of Theorem 1, we will formulate some useful
statements.

It is well known that Besov spaces are isomorphic to some sequence spaces
(Lizorkin, Triebel, Meyer). Let us describe this isomorphism in the case of
spaces on $\mathbb{R}$. Let $\psi$, $\varphi$ be a wavelet function and a
scaling function, correspondingly. We will make use of regular Daubechies
wavelets with compact support (see. [3], §1.2): $\psi,\varphi\in
C^{r_{0}}(\mathbb{R})$ and $\int\psi(x)x^{j}\,dx=0$, $0\leqslant j<r_{0}$;
with some fixed $r_{0}\in\mathbb{N}$.

Denote $\psi_{k,j}(x):=2^{k/2}\psi(2^{k}x-j)$,
$\varphi_{0,j}(x):=\varphi(x-j)$, $k,j\in\mathbb{Z}$.

We consider the space $\ell^{\sigma}_{p,\theta}$ of sequences
$(\lambda_{k,j})_{k\geqslant 0,j\in\mathbb{Z}}$ with the norm

$\|(\lambda_{k,j})_{k\geqslant
0,j\in\mathbb{Z}}\|_{\ell^{\sigma}_{p,\theta}}=\begin{cases}\left(\sum_{k\geqslant
0}2^{k\sigma\theta}\|(\lambda_{k,j})_{j\in\mathbb{Z}}\|_{\ell_{p}}^{\theta}\right)^{1/\theta},&\quad\theta<\infty,\\\
\sup_{k\geqslant
0}2^{k\sigma}\|(\lambda_{k,j})_{j\in\mathbb{Z}}\|_{\ell_{p}},&\quad\theta=\infty.\end{cases}$

######  Theorem A (See. [3], §1.2, Theorem 1.20).

Suppose $0<r<r_{0}$; $p,\theta\in[1,\infty]$. Then $f\in L_{p}(\mathbb{R})$
lies in $\mathcal{B}^{r}_{p,\theta}(\mathbb{R})$ if and only if the following
norm is finite:

$\|f\|:=\|(\langle
f,\varphi_{0,j}\rangle)_{j\in\mathbb{Z}}\|_{\ell_{p}}+\|(\langle
f,\psi_{k,j}\rangle)_{k\geqslant
0,j\in\mathbb{Z}}\|_{\ell^{\sigma}_{p,\theta}},\;\sigma:=r+\frac{1}{2}-\frac{1}{p}.$
(6)

This norm is equivalent to the norm of
$\mathcal{B}^{r}_{p,\theta}(\mathbb{R})$.

See also [7], Th.1.13.

We will use later that Besov classes grow as $\theta$ increases
($B^{1}_{1,\theta_{1}}$ is embedded in $B^{1}_{1,\theta_{2}}$ as
$\theta_{1}<\theta_{2}$), it is easily seen from this theorem.

We should discretize the norm, too. Consider the subspace in $L_{p}$:

$L_{p}\langle\Psi_{+}\rangle:=\overline{\mathrm{span}}\,\Psi_{+},\quad\Psi_{+}:=\\{\psi_{k,j}\colon
k\geqslant 0,\,j\in\mathbb{Z}\\}.$

(the closure is in $L_{p}(\mathbb{R})$).

######  Lemma 3.

Let $f\in L_{p}\langle\Psi_{+}\rangle$. Then for $2\leqslant p<\infty$ we have

$\|(\langle f,\psi_{k,j}\rangle)_{k\geqslant
0,j\in\mathbb{Z}}\|_{\ell^{1/2-1/p}_{p,p}}\ll\|f\|_{p}\ll\|(\langle
f,\psi_{k,j}\rangle)_{k\geqslant 0,j\in\mathbb{Z}}\|_{\ell^{1/2-1/p}_{p,2}}.$
(7)

We provide the proof for completeness; it is possible that such bounds are
known. Similar inequalities had appeared in the context of Kolmogorov widths
in, e.g., [9] for Haar wavelets.

###### Proof.

We will make use of the following facts about wavelets.

  * (i)

Wavelets $\\{\psi_{k,j}\\}$ are an unconditional basis in $L_{p}(\mathbb{R})$,
see. [2], §8.2; hence, using [2], §7.3, Corrolary 7.11, we have

$\|f\|_{p}\asymp\|(\sum_{k,j}\langle
f,\psi_{k,j}\rangle^{2}|\psi_{k,j}|^{2})^{1/2}\|_{p}.$

  * (ii)

For fixed $k$, inequality [2], §8.1, Proposition 8.3 gives

$\|\sum_{j\in\mathbb{Z}}\langle f,\psi_{k,j}\rangle\psi_{k,j}\|_{p}\asymp
2^{k(1/2-1/p)}\|(\langle f,\psi_{k,j}\rangle)_{j}\|_{\ell_{p}}.$

Use (i), the inequality $(a+b)^{r}\geqslant a^{r}+b^{r}$ for $r\geqslant 1$,
$a,b\geqslant 0$, and the relation $\|\psi_{k,j}\|_{p}\asymp 2^{k(1/2-1/p)}$:

$\|f\|_{p}^{p}\asymp\int|\sum_{k\geqslant 0,j\in\mathbb{Z}}\langle
f,\psi_{k,j}\rangle^{2}\psi_{k,j}(t)^{2}|^{p/2}\,dt\geqslant\sum_{k\geqslant
0,j\in\mathbb{Z}}\|\langle f,\psi_{k,j}\rangle\psi_{k,j}\|_{p}^{p}\asymp\\\
\asymp\sum_{k\geqslant 0}2^{kp(1/2-1/p)}\|(\langle
f,\psi_{k,j}\rangle)_{j}\|_{\ell_{p}}^{p}.$

It gives us the required lower bound.

For the upper bound we use (i), put $g_{k}=(\sum_{j\in\mathbb{Z}}\langle
f,\psi_{k,j}\rangle^{2}\psi_{k,j}^{2})^{1/2}$ and obtain

$\|f\|_{p}^{2}\asymp\|(\sum_{k\geqslant
0}g_{k}^{2})^{1/2}\|_{p}^{2}=\|\sum_{k\geqslant
0}g_{k}^{2}\|_{p/2}\leqslant\sum_{k\geqslant
0}\|g_{k}^{2}\|_{p/2}=\sum_{k\geqslant 0}\|g_{k}\|_{p}^{2}.$

Now, using (i) and then (ii),

$\|g_{k}\|_{p}=\|(\sum_{j\in\mathbb{Z}}\langle
f,\psi_{k,j}\rangle^{2}\psi_{k,j}^{2})^{1/2}\|_{p}\asymp\\\
\asymp\|\sum_{j\in\mathbb{Z}}\langle f,\psi_{k,j}\rangle\psi_{k,j}\|_{p}\asymp
2^{k(1/2-1/p)}\|(\langle f,\psi_{k,j}\rangle)_{j}\|_{\ell_{p}}.$

Therefore,

$\|f\|_{p}^{2}\ll\sum_{k\geqslant 0}2^{2k(1/2-1/p)}\|(\langle
f,\psi_{k,j}\rangle)_{j}\|_{\ell_{p}}^{2},$

as required. Lemma is proved. ∎

###### Proof of Theorem 1.

We will use Theorem A and Lemma 3 to discretize our problem and then we will
bound widths using Theorem 2.

Discretization. It is convenient to change coordinates (to get octahedra):

$x_{k,j}:=2^{k/2}\langle f,\psi_{k,j}\rangle.$

Moreover, as we work with function classes on the segment, we may restrict
indices to the set $T:=\\{(k,j)\colon k\geqslant 0,\,j=1,\ldots,2^{k}\\}$.
Consider the subspace $L_{q}\langle\Psi_{T}\rangle$, spanned by
$\Psi_{T}:=\\{\psi_{k,j}\colon(k,j)\in T\\}$. The support of functions from
$\Psi_{T}$ is contained in some segment $[-a,a]$; this is also true for
$L_{q}\langle\Psi_{T}\rangle$. The natural projector $L_{q}(\mathbb{R})\to
L_{q}\langle\Psi_{T}\rangle$ is bounded (because wavelets form a basis), hence

$d_{n}(B^{1}_{1,\theta}[-a,a],L_{q}[-a,a])\gg
d_{n}(B^{1}_{1,\theta}(\mathbb{R})\cap
L_{q}\langle\Psi_{T}\rangle,L_{q}\langle\Psi_{T}\rangle).$ (8)

######  Remark.

One could use periodic wavelets in
$\mathcal{B}^{r}_{p,\theta}(\mathbb{R}/\mathbb{Z})$ instead, see [3], §1.3,
Theorem 1.37; however, this also brings technical complications. The most
convenient for us would be Haar wavelets, but they do not form an
unconditional basis for $\mathcal{B}^{1}_{1,\theta}$, see [17].

We denote by $\ell^{\sigma}_{p,\theta}(T)$ the analog of the space
$\ell^{\sigma}_{p,\theta}$ for vectors $x\in\mathbb{R}^{T}$:

$\|x\|_{\ell^{\sigma}_{p,\theta}(T)}:=\begin{cases}\left\\{\sum_{k\geqslant
0}2^{k\sigma\theta}\|(x_{k,j})_{1\leqslant j\leqslant
2^{k}}\|_{\ell_{p}}^{\theta}\right\\}^{1/\theta},&\quad\theta<\infty,\\\
\sup_{k\geqslant 0}2^{k\sigma}\|(x_{k,j})_{1\leqslant j\leqslant
2^{k}}\|_{\ell_{p}},&\quad\theta=\infty,\end{cases}$

$b^{\sigma}_{p,\theta}(T)$ is the unit ball of this space.

Theorem A gives that for $f\in L_{q}\langle\Psi_{T}\rangle$ we have

$\|f\|_{\mathcal{B}^{1}_{1,\theta}}\asymp\|\langle
f,\psi_{k,j}\rangle\|_{\ell^{1/2}_{1,\theta}(T)}=\|x\|_{\ell^{0}_{1,\theta}(T)},\quad
x_{k,j}=2^{k/2}\langle f,\psi_{k,j}\rangle.$ (9)

Indeed, the first term in (6) is zero as $\psi_{k,j}\perp\varphi_{0,i}$,
$k\geqslant 0$ (this follows from the definition of wavelets and scaling
functions). So, we have proved that the set $b^{0}_{1,\theta}(T)$ is the
discretization of $B^{1}_{1,\theta}\cap L_{q}\langle\Psi_{T}\rangle$. The set
$b^{0}_{1,\theta}(T)$ is the infinite-dimensional octahedron for $\theta=1$
and the cartesian product of octahedra of dimensions $1,2,\ldots,2^{k},\ldots$
for $\theta=\infty$.

Let us write the lower bound from (7) for $p=q$, $f\in
L_{q}\langle\Psi_{T}\rangle$, in the coordinates $x_{k,j}$:

$\|f\|_{q}^{q}\gg\sum_{k=0}^{\infty}2^{-k}\sum_{j=1}^{2^{k}}|x_{k,j}|^{q}=\|x\|^{q}_{\ell^{-1/q}_{q,q}(T)}.$
(10)

It is important to note that as $q>2$, we have $\|f\|_{q}\geqslant\|f\|_{2}$
and hence $\|f\|_{q}\geqslant\|x\|_{\ell^{-1/2}_{2,2}(T)}$. Using (8), (10)
and (9), we obtain the “lower” discretization:

$d_{n}(B^{1}_{1,\theta}[-a,a],L_{q}[-a,a])\gg
d_{n}(b^{0}_{1,\theta}(T),\ell^{-1/q}_{q,q}(T)\cap\ell^{-1/2}_{2,2}(T)).$ (11)

(The change of segment from $[0,1]$ to $[-a,a]$ makes no difference.)

The “upper” discretization is analogous. Pick $f\in B^{1}_{1,\theta}[0,1]$. We
can extend $f$ from the segment to the whole line and control it’s norm and
support. To do this, one can extend $f$ to $\mathbb{R}$ (see the Remark after
Besov space definition) and then multiply it by some “cut function”, see [5],
Lemma 3. We obtain $f\colon\mathbb{R}\to\mathbb{R}$ with support $[-b,b]$.
Only finite number of functions $\varphi_{0,j}$ have support intersecting
$[-b,b]$. We subtract from $f$ the linear combination of such functions (it is
an element of $n_{0}$-dimensional space for some fixed $n_{0}$) and get a
function with support in larger segment $[-B,B]$, but orthogonal to
$\psi_{k,j}$, $k<0$ (orthogonality follows from the wavelet definition):

$\widetilde{f}:=f-\sum\langle\varphi_{0,j},f\rangle\varphi_{0,j}\in
L_{q}\langle\Psi_{+}\rangle,\;\operatorname{supp}\widetilde{f}\subset[-B,B],\;\|\widetilde{f}\|_{\mathcal{B}^{1}_{1,\theta}(\mathbb{R})}\ll\|f\|_{\mathcal{B}^{1}_{1,\theta}[0,1]}.$

Denote

$T^{\prime}:=\\{(k,j)\colon k\geqslant
0,\,\operatorname{supp}\psi_{k,j}\cap[-B,B]\neq\varnothing\\},\quad\Psi_{T^{\prime}}:=\\{\psi_{k,j}\colon(k,j)\in
T^{\prime}\\},$

then $\widetilde{f}\in L_{q}\langle\Psi_{T^{\prime}}\rangle$. Therefore

$d_{n+n_{0}}(B^{1}_{1,\theta}[0,1],L_{q}[0,1])\ll
d_{n}(B^{1}_{1,\theta}(\mathbb{R})\cap
L_{q}\langle\Psi_{T^{\prime}}\rangle,L_{q}).$

The inequality (7) in the coordinates $x_{k,j}$ becomes

$\|f\|_{q}^{2}\ll\sum_{k\geqslant
0}\left(2^{-k}\sum_{j=1}^{2^{k}}|x_{k,j}|^{q}\right)^{2/q}=\|x\|^{2}_{\ell^{-1/q}_{q,2}(T)},$

and, finally,

$d_{n+n_{0}}(B^{1}_{1,\theta}[0,1],L_{q}[0,1])\ll
d_{n}(b^{0}_{1,\theta}(T^{\prime}),\ell^{-1/q}_{q,2}(T^{\prime}))\asymp
d_{n}(b^{0}_{1,\theta}(T),\ell^{-1/q}_{q,2}(T)).$ (12)

Lower bound. We will bound (11), using theorem 2 in the form (1). Obviously,
$\ell^{-1/q}_{q,q}(T)$ and $\ell^{-1/2}_{2,2}(T)$ are weighted norms with
weights $w_{k,i}=2^{-k}$.

Let us start with the case $\theta=1$; the set $b^{0}_{1,1}(T)$ is the
octahedron. We denote by $k_{0}$ and $k_{1}$ the minimal and maximal $k$,
correspondingly, that satisfy the condition $4n<2^{k}<n^{\tilde{q}/2}$, with
some fixed $\tilde{q}\in(2,q)$, e.g. $\tilde{q}=1+q/2$. We apply Theorem 2 and
(1) in the following situation:

  * •

from $x\in\mathbb{R}^{T}$ we take $N=\sum_{k=k_{0}}^{k_{1}}2^{k}$ coordinates
$x_{k,i}$, $k_{0}\leqslant k\leqslant k_{1}$, $i=1,\ldots,2^{k}$; the set
$b^{0}_{1,1}(T)$ becomes $B_{1}^{N}$;

  * •

weights $w_{k,i}=2^{-k}$, $k_{0}\leqslant k\leqslant k_{1}$,
$i=1,\ldots,2^{k}$;

  * •

$m=1$ (one block);

  * •

$q_{1}=2$, $q_{2}=q$.

Let us check that the conditions of Theorem 2 are met. Condition 1):

$\max w_{k,i}=2^{-k_{0}}<(4n)^{-1},\quad\sum w_{k,i}=(k_{1}-k_{0}+1)\asymp\log
n.$

Condition 2) is satisfied for large enough $n$. Hence from (1) we have the
bound

$d_{n}\gg\min(n^{-1/2}(\log n)^{1/2},2^{-k_{1}/q})\asymp n^{-1/2}\log^{1/2}n,$

as $2^{-k_{1}/q}>n^{-\frac{1}{2}\tilde{q}/q}$.

The case $\theta\in(1,2]$ follows from $\theta=1$ as our class grows when
$\theta$ increases.

In the case $\theta=\infty$ we apply Theorem 2 for the same coordinates and
weights, but now the set $b^{0}_{1,\infty}(T)$ restricted to $N$ coordinates
is the product of octahedra $\prod_{k=k_{0}}^{k_{1}}B_{1}^{2^{k}}$ and there
are $m\asymp\log n$ blocks $\Delta_{s}=\\{w_{k,i}\colon k=s\\}$. Arguing as
before, we obtain

$d_{n}\gg\min(\sqrt{\frac{\log n}{n}}(\log n)^{1/2},2^{-k_{1}/q})\asymp
n^{-1/2}\log n.$

We have proven slightly more: if we cut $T$ by $m\asymp\log n$ levels, i.e.,
replace it with

$T_{m}:=\\{(k,j)\colon 0\leqslant k<m,\;j=1,\ldots,2^{k}\\},$

then the bound is valid:

$d_{n}(b^{0}_{1,\infty}(T_{m}),\ell^{-1/q}_{q,q}(T_{m})\cap\ell^{-1/2}_{2,2}(T_{m}))\gg
n^{-1/2}\log n,\quad m\asymp\log n.$ (13)

Finally, $\theta\in(2,\infty)$. Use (11) with $T$ replaced by $T_{m}$,
$m\asymp\log n$. Embed $b^{0}_{1,\infty}(T_{m})\subset
m^{1/\theta}b^{0}_{1,\theta}(T_{m})$ and use (13) to get the required bound.

Upper bound. First we prove that in (12) it is enough to consider $m\asymp\log
n$ blocks. Denote $x[k]:=(x_{k,j})_{j=1}^{2^{k}}$. If $x\in
b^{0}_{1,\theta}(T)$, then $\|x[k]\|_{1}\leqslant 1$ for all $k$; hence,
$\|x[k]\|_{q}\leqslant 1$,

$\sum_{k\geqslant m}(2^{-k}\|x[k]\|_{q}^{q})^{2/q}\ll 2^{-2m/q}.$

For $m\geqslant C\log n$ this “tail” is less than $n^{-C_{1}}$; it has no
effect on the width for large $C_{1}$. So, it is enought to bound
$d_{n}(b^{0}_{1,\theta}(T_{m}),\ell^{-1/q}_{q,2}(T_{m}))$, $m\asymp\log n$.

We start with the case $\theta=2$. The set of the coordinates is split to
blocks $\Delta_{k}=\\{(k,j)\\}_{j=1}^{2^{k}}$. To approximate in blocks of
size $2^{k}<n/4$ we take the whole space $\mathbb{R}^{\Delta_{k}}$, it
requires about a half of the dimension ($\leqslant n/2$). In other blocks we
take the subspaces of dimension $l\approx n/(2m)$ which are extremal for the
widths $d_{l}(B_{1}^{2^{k}},\ell_{q}^{2^{k}})\asymp l^{-1/2}2^{k/q}$, as in
(2). We obtain the approximation

$\|x[k]-y[k]\|_{q}\ll(n/m)^{-1/2}2^{k/q}\|x[k]\|_{1},$

so

$(2^{-k}\sum_{j=1}^{2^{k}}|x_{k,j}-y_{k,j}|^{q})^{2/q}\ll(n/m)^{-1}\|x[k]\|_{1}^{2}.$

Let us sum over $k$:

$\|x-y\|_{\ell^{-1/q}_{q,2}(T_{m})}^{2}\ll(n/m)^{-1}\sum_{k=0}^{m-1}\|x[k]\|_{1}^{2}\leqslant(n/m)^{-1},$

as $x\in b^{0}_{1,2}(T_{m})$. So, the error of the approximation is bounded by
$Cn^{-1/2}\log^{1/2}n$.

The case $\theta\in[1,2)$ follows from $\theta=2$ as the class grows as
$\theta$ increases.

If $\theta\geqslant 2$, we use that $b^{0}_{1,\theta}(T_{m})\subset
m^{1/2-1/\theta}b^{0}_{1,2}(T_{m})$:

$d_{n}(b^{0}_{1,\theta}(T_{m}),\ell^{-1/q}_{q,2}(T_{m}))\leqslant
m^{1/2-1/\theta}d_{n}(b^{0}_{1,2}(T_{m}),\ell^{-1/q}_{q,2}(T_{m}))\ll
n^{-1/2}\log^{1-1/\theta}n.$

∎

##  3 Future work

It would be interesting to research other classes related to $W^{1}_{1}$,
e.g., Lizorkin–Triebel classes $F^{1}_{1,\theta}$.

It is possible that one can apply Theorem 2 for widths of classes of functions
of several variables.

Author thanks reviewer for useful commens on Theorem 2 and A.I. Tyulenev for
pointing out paper [5].

## References

  * [1] B.S. Kashin, “ Some properties of matrices of bounded operators from the space $\ell_{2}^{n}$ into $\ell_{2}^{m}$” (Russian), Izv. Akad. Nauk Armyan. SSR Ser. Mat., 15:5 (1980), 379–394.
  * [2] P. Wojtaszczyk, “A Matchematical Introduction to Wavelets”, Cambridge University Press, 1997.
  * [3] H. Triebel, Function Spaces and Wavelets on Domains, EMS Tracts in Mathematics, 2008.
  * [4] H. Triebel, Theory of Function Spaces III, Birkhäuser Verlag, 2006.
  * [5] O.V. Besov, “Continuation of certain classes of differentiable functions beyond the boundary of a region” (Russian), Trudy Mat. Inst. Steklov., 77 (1965), 35–44.
  * [6] R. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, 2003.
  * [7] Dorothee Haroske, L.Skrzypczak, “Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I”, Rev. Mat. Complut., 21:1 (2008), 135–177.
  * [8] D. Dung, V. Temlyakov, T. Ullrich, Hyperbolic Cross Approximation Springer, 2018.
  * [9] E.D. Kulanin, “On the diameters of a class of functions of bounded variation in the space $L_{q}(0,1)$, $2<q<\infty$”, Russian Mathematical Surveys 38:5 (1983), 146–147.
  * [10] G.G. Lorentz, M. Golitschek, Y. Makovoz, Constructive Approximation: Advanced Problems, 1996\.
  * [11] R.S. Ismagilov, “On $n$-dimensional diameters of compacts in a Hilbert space”, Functional Analysis and Its Applications, 2 (1968), 125–132.
  * [12] E.S. Belinskii, “Approximation of periodic functions by “floating” system of exponentials and trigonometric widths” (Russian), Yaroslavl, 1984.
  * [13] E.M. Galeev, “Kolmogorov widths of classes of periodic functions of one and several variables”, Mathematics of the USSR-Izvestiya, 36:2 (1991), 435–448.
  * [14] Dinh Dung, “Approximation by trigonometric polynomials of functions of several variables on the torus” Math. USSR-Sb., 59:1 (1988), 247–267.
  * [15] A. Romanyuk, “Approximation of Besov classes of multivariate periodic functions in the space $L_{q}$”, Ukranian Math. J., 43 (1991), 1398–1408.
  * [16] E.M. Galeev, “Widths of the Besov Classes $B^{r}_{p,\theta}(\mathbb{T}^{d})$, Math. Notes, 69:5 (2001), 605–613.
  * [17] G. Garrigós, A. Seeger, T. Ullrich, “Basis properties of the Haar system in limiting Besov spaces”, arXiv:1901.09117.
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  * [19] Yu.V. Malykhin, K.S. Ryutin,‘ “The Product of Octahedra is Badly Approximated in the $\ell_{2,1}$-Metric”, Math. Notes, 101:1 (2017), 94–99.


# Comments on the Bell-Clauser-Horne-Shimony-Holt inequality

S. P. Sorella

UERJ – State University of Rio de Janeiro,
Physics Institute – Department of Theoretical Physics – Rua São Francisco
Xavier 524,
20550-013, Maracanã, Rio de Janeiro, Brazil
<EMAIL_ADDRESS>
###### Abstract

We discuss the relationship between the Bogoliubov transformations, squeezed
states, entanglement and maximum violation of the Bell-CHSH inequality. In
particular, we point out that the construction of the four bounded operators
entering the Bell-CHSH inequality can be worked out in a simple and general
way, covering a large variety of models, ranging from Quantum Mechanism to
relativistic Quantum Field Theories. Various examples are employed to
illustrate the above mentioned framework. We start by considering a pair of
entangled spin 1 particles and a squeezed oscillator in Quantum Mechanics,
moving then to the relativistic complex quantum scalar field and to the
analysis of the vacuum state in Minkowski space-time in terms of the left and
right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to
be parametrized by the Unruh temperature.

##  1 Introduction

The aim of this work is that of pointing out that the construction of the four
Hermitian operators $A_{i}$, $B_{i}$, $i=1,2$

$A^{2}_{i}=B^{2}_{i}=1\;,\qquad[A_{i},B_{k}]=0\;.$ (1)

characterizing the quantum violation of the Bell-Clauser-Horne-Shimony-Holt
inequality [1, 2, 3], i.e.

$|\langle\psi|{\cal
C}_{CHSH}|\psi\rangle|=|\langle\psi|(A_{1}+A_{2})B_{1}+(A_{1}-A_{2})B_{2}|\psi\rangle|>2\;,$
(2)

can be achieved in a rather simple and elegant way, which turns out to have
general applicability, covering models ranging from Quantum Mechanics to more
sophisticated examples such as: relativistic Quantum Field Theories.

In order to illustrate how the setup works, we proceed first by discussing two
examples from Quantum Mechanics.

##  2 Entangled pair of spin 1 particles

Let us start by considering a pair of entangled spin 1 particles. As entangled
state $|\psi_{s}\rangle$, we take the singlet state:

$|\psi_{s}\rangle=\left(\frac{|1\rangle_{A}|-1\rangle_{B}-|0\rangle_{A}|0\rangle_{B}+|-1\rangle_{A}|1\rangle_{B}}{\sqrt{3}}\right)\;.$
(3)

It is easily checked that expression (3) can be thought as the vacuum state of
the spin Hamiltonian

$H={\vec{S}}_{A}\cdot{\vec{S}}_{B}=\frac{1}{2}\left({\vec{S}}_{A}+{\vec{S}}_{B}\right)^{2}-2\;.$
(4)

with $\left({\vec{S}}_{A},{\vec{S}}_{B}\right)$ denoting the spin 1 matrices.
For the operators $(A_{i},B_{i})$ we write

$\displaystyle A_{i}|-1\rangle_{A}$ $\displaystyle=$ $\displaystyle
e^{i\alpha_{i}}|0\rangle_{A}\;,\qquad
A_{i}|0\rangle_{A}=e^{-i\alpha_{i}}|-1\rangle_{A}\;,\qquad
A_{i}|1\rangle_{A}=|1\rangle_{A}\;,$ $\displaystyle B_{i}|1\rangle_{B}$
$\displaystyle=$ $\displaystyle e^{i\beta_{i}}|0\rangle_{B}\;,\qquad
B_{i}|0\rangle_{B}=e^{-i\beta_{i}}|1\rangle_{B}\;,\qquad
B_{i}|-1\rangle_{B}=|-1\rangle_{B}\;,$ (5)

where $(\alpha_{i},\beta_{i})$ are arbitrary real coefficients. The Hermitian
operators $A_{i}$, $B_{i}$, $i=1,2$ fulfill the requirement (1).

For the Bell-CHSH correlator we obtain

$\langle\psi_{s}|{\cal
C}_{CHSH}|\psi_{s}\rangle=\frac{2}{3}\left(1-cos(\alpha_{1}+\beta_{1})-cos(\alpha_{2}+\beta_{1})-cos(\alpha_{1}+\beta_{2})+cos(\alpha_{2}+\beta_{2})\right)\;.$
(6)

Choosing, for example, $\alpha_{2}=\beta_{2}=0$, $\alpha_{1}=\frac{\pi}{2}$,
$\beta_{1}=\frac{3\pi}{4}$, one gets the violation

$|\langle\Omega|{\cal C}_{CHSH}|\Omega\rangle|=\frac{2(2+\sqrt{2})}{3}\approx
2.27\;,$ (7)

which compares well with the value reported by [4].

The same reasoning applies to the Bell spin $1/2$ singlet state

$|\psi_{s}\rangle=\left(\frac{|+\rangle_{A}|-\rangle_{B}-|-\rangle_{A}|+\rangle_{B}}{\sqrt{2}}\right)\;.$
(8)

For the operators $(A_{i},B_{i})$ we have now

$\displaystyle A_{i}|+\rangle_{A}$ $\displaystyle=$ $\displaystyle
e^{i\alpha_{i}}|-\rangle_{A}\;,\qquad
A_{i}|-\rangle_{A}=e^{-i\alpha_{i}}|+\rangle_{A}\;,\;,$ $\displaystyle
B_{i}|+\rangle_{B}$ $\displaystyle=$ $\displaystyle
e^{i\beta_{i}}|-\rangle_{B}\;,\qquad
B_{i}|-\rangle_{B}=e^{-i\beta_{i}}|+\rangle_{B}\;.$ (9)

Setting $\alpha_{1}=0$, $\alpha_{2}=\frac{\pi}{2}$, $\beta_{1}=\frac{\pi}{4}$,
$\beta_{2}=-\frac{\pi}{4}$, one recovers Tsirelson’s bound [3], namely

$|\langle\psi_{s}|{\cal C}_{CHSH}|\psi_{s}\rangle|=2\sqrt{2}\;.$ (10)

##  3 Two degrees of freedom squeezed oscillator

In this second example we consider a two degrees of freedom squeezed
oscillator. Let us begin by introducing two annihilation operators $(a,b)$
satisfying

$\displaystyle\left[a,a^{\dagger}\right]$ $\displaystyle=$ $\displaystyle
1\;,\qquad[a^{\dagger},a^{\dagger}]=0\;,\qquad[a,a]=0\;,$
$\displaystyle\left[b,b^{\dagger}\right]$ $\displaystyle=$ $\displaystyle
1\;,\qquad[b^{\dagger},b^{\dagger}]=0\;,\qquad[b,b]=0\;,$
$\displaystyle\left[a,b\right]$ $\displaystyle=$ $\displaystyle
0\;,\qquad[a,b^{\dagger}]=0\;.$ (11)

The operators $(a,b)$ annihilate the state $|0\rangle$, which is not the
vacuum state of the system, i.e.

$a|0\rangle=b|0\rangle=0\;.$ (12)

Let us introduce the new operators $(\alpha,\beta)$, obtained through the
Bogoliubov transformations

$\alpha=\frac{(a-\eta
b^{\dagger})}{\sqrt{1-\eta^{2}}}\;,\qquad\beta=\frac{(b-\eta
a^{\dagger})}{\sqrt{1-\eta^{2}}}\;,$ (13)

where $\eta$ is a real parameter, $0<\eta<1$.

It is easily checked that the operators $(\alpha,\beta)$ fulfill the same
commutation relations of eq.(11), namely

$\displaystyle\left[\alpha,\alpha^{\dagger}\right]$ $\displaystyle=$
$\displaystyle
1\;,\qquad[\alpha^{\dagger},\alpha^{\dagger}]=0\;,\qquad[\alpha,\alpha]=0\;,$
$\displaystyle\left[\beta,\beta^{\dagger}\right]$ $\displaystyle=$
$\displaystyle
1\;,\qquad[\beta^{\dagger},\beta^{\dagger}]=0\;,\qquad[\beta,\beta]=0\;,$
$\displaystyle\left[\alpha,\beta\right]$ $\displaystyle=$ $\displaystyle
0\;,\qquad[\alpha,\beta^{\dagger}]=0\;.$ (14)

We consider now the normalized squeezed state

$|\eta\rangle=\sqrt{(1-\eta^{2})}\;e^{\eta
a^{\dagger}b^{\dagger}}|0\rangle\;,\qquad\langle\eta|\eta\rangle=1\;.$ (15)

Let us show that the state $|\eta\rangle$ is annihilated by the Bogoliubov
operators $(\alpha,\beta)$:

$\alpha|\eta\rangle=\beta|\eta\rangle=0\;.$ (16)

In fact, we have

$\displaystyle(a-\eta b^{\dagger})e^{\eta a^{\dagger}b^{\dagger}}|0\rangle$
$\displaystyle=$ $\displaystyle\left[a,e^{\eta
a^{\dagger}b^{\dagger}}\right]|0\rangle-\eta b^{\dagger}e^{\eta
a^{\dagger}b^{\dagger}}|0\rangle$ (17) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{\eta^{n}}{n!}(b^{\dagger})^{n}\left[a,(a^{\dagger})^{n}\right]|0\rangle-\eta
b^{\dagger}e^{\eta a^{\dagger}b^{\dagger}}|0\rangle$ $\displaystyle=$
$\displaystyle\eta
b^{\dagger}\sum_{n=1}^{\infty}\frac{\eta^{n-1}}{(n-1)!}(b^{\dagger})^{(n-1)}(a^{\dagger})^{(n-1)}|0\rangle-\eta
b^{\dagger}e^{\eta a^{\dagger}b^{\dagger}}|0\rangle=0\;.$

Similarly for the operator $\beta$. As a consequence, the squeezed state
$|\eta\rangle$ turns out to be the vacuum state of the Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle\alpha^{\dagger}\alpha+\beta^{\dagger}\beta=\frac{(1+\eta^{2})}{(1-\eta^{2})}\left(a^{\dagger}a+b^{\dagger}b\right)-\frac{2\eta}{(1-\eta^{2})}\left(a^{\dagger}b^{\dagger}+ab\right)+2\eta^{2}.$
$\displaystyle H|\eta\rangle$ $\displaystyle=$ $\displaystyle 0\;.$ (18)

This expression is nothing but the Hamiltonian worked out in [5] in order to
establish the violation of the Bell-CHSH inequality in relativistic free
Quantum Field Theory. It becomes apparent now that the role of the parameter
$\eta$, $0<\eta<1$, is that of a coupling constant.

###  3.1 Maximum violation of the Bell-CHSH inequality

We are ready now to discuss the violation of the Bell-CHSH exhibited by this
model. To introduce the four Hermitian operators $A_{i}$, $B_{i}$, $i=1,2$

$A^{2}_{i}=B^{2}_{i}=1\;,\qquad[A_{i},B_{k}]=0\;,$ (19)

we proceed in exactly the same way as done in the case of the spin 1 example.
We write the squeezed state $|\eta\rangle$ as

$\displaystyle|\eta\rangle$ $\displaystyle=$
$\displaystyle\sqrt{(1-\eta^{2})}\;\sum_{n=0}^{\infty}\eta^{n}\frac{(a^{\dagger}b^{\dagger})^{n}}{n!}|0\rangle$
(20) $\displaystyle=$
$\displaystyle\sqrt{(1-\eta^{2})}\;\sum_{n=0}^{\infty}\left(\eta^{(2n)}|(2n_{a})(2n_{b})\rangle+\eta^{(2n+1)}|(2n_{a}+1)(2n_{b}+1)\rangle\right)\;,$

where $|m_{a}m_{b}\rangle$ stands for the normalized state

$|m_{a}m_{b}\rangle=\frac{(a^{\dagger})^{m}(b^{\dagger})^{m}}{m!}|0\rangle\;.$
(21)

For the operators $A_{i}$, $B_{i}$, $i=1,2$, we have

$\displaystyle A_{i}|(2n_{a})\;m_{b}\rangle$ $\displaystyle=$ $\displaystyle
e^{i\alpha_{i}}|(2n_{a}+1)\;m_{b}\rangle\;,\qquad
A_{i}|(2n_{a}+1)\;m_{b}\rangle=e^{-i\alpha_{i}}|(2n_{a})\;m_{b}\rangle\;,$
$\displaystyle B_{i}|m_{a}\;(2n_{b})\rangle$ $\displaystyle=$ $\displaystyle
e^{i\beta_{i}}|m_{a}\;(2n_{b}+1)\rangle\;,\qquad
B_{i}|m_{a}\;(2n_{b}+1)\rangle=e^{-i\beta_{i}}|m_{a}\;(2n_{b})\rangle\;.$ (22)

The operator $A_{i}$ acts only on the first entry, while $B_{i}$ only on the
second one.

A quick calculation gives

$\langle\eta|\;A_{k}B_{i}\;|\eta\rangle=\frac{2\eta}{1+\eta^{2}}\;\cos(\alpha_{k}+\beta_{i})\;.$
(23)

Therefore, for the Bell-CHSH correlator, one finds

$\displaystyle\langle\eta|{\cal C}_{CHSH}|\eta\rangle$ $\displaystyle=$
$\displaystyle\langle\eta|(A_{1}+A_{2})B_{1}+(A_{1}-A_{2})B_{2}|\eta\rangle$
(24) $\displaystyle=$
$\displaystyle\frac{2\eta}{1+\eta^{2}}\left(\cos(\alpha_{1}+\beta_{1})+\cos(\alpha_{2}+\beta_{1})+\cos(\alpha_{1}+\beta_{2})-\cos(\alpha_{2}+\beta_{2})\right)\;.$

Setting [5]

$\alpha_{1}=0\;,\qquad\beta_{1}=-\frac{\pi}{4}\;,\qquad\alpha_{2}=\frac{\pi}{2}\;,\qquad\beta_{2}=\frac{\pi}{4}\;,$
(25)

expression (24) becomes

$\langle\eta|{\cal
C}_{CHSH}|\eta\rangle=2\;\frac{2\sqrt{2}\eta}{1+\eta^{2}}\;,$ (26)

implying in a violation of the Bell-CHSH inequality whenever

$\sqrt{2}-1<\eta<1\;.$ (27)

In particular, the maximum violation is attained for values of $\eta\approx
1$, namely

$\langle\eta|{\cal C}_{CHSH}|\eta\rangle\approx 2\sqrt{2}\;.$ (28)

##  4 The relativistic complex scalar quantum Klein-Gordon field

Let us move now to Quantum Field Theory, by considering the example of a free
complex massive scalar Klein-Gordon field, i.e.

${\cal
L}=\left(\partial^{\mu}\varphi^{\dagger}\partial_{\mu}\varphi-m^{2}\varphi^{\dagger}\varphi\right)\;.$
(29)

Expanding $\varphi$ in terms of annihiliation and creation operators, one gets

$\varphi(t,{\vec{x}})=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega_{k}}\left(e^{-ikx}a_{k}+e^{ikx}b^{\dagger}_{k}\right)\;,\qquad
k^{0}=\omega_{k}=\sqrt{{\vec{k}}^{2}+m^{2}}\;,$ (30)

where

$[a_{k},a^{\dagger}_{q}]=[b_{k},b^{\dagger}_{q}]=(2\pi)^{3}2\omega_{k}\delta^{3}({\vec{k}-\vec{q}})\;,$
(31)

are the non vanishing canonical commutation relations. Expression (30) is a
too singular object, being in fact an operator valued distribution [6]. In
order to give a well defined meaning to eq.(30), one introduces the smeared
operators

$a_{f}=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega_{k}}{\hat{f}}(\omega_{k},{\vec{k}})\;a_{k}\;,\qquad
b_{g}=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega_{k}}{\hat{g}}(\omega_{k},{\vec{k}})\;b_{k}\;,$
(32)

with

${\hat{f}}(p)=\int d^{4}x\;e^{-ipx}f(x)\;,\qquad{\hat{g}}(p)=\int
d^{4}x\;e^{-ipx}g(x).\;,$ (33)

where $(f(x),g(x))$ are test functions belonging to the space of compactly
supported smooth functions ${\cal C}_{0}^{\infty}(\mathbb{R}^{4})$. The
support of $f(x)$, $supp_{h}$, is the region in which the test function $f(x)$
is non-vanishing.

When rewritten in terms of the operators $(a_{f},b_{g})$, the canonical
commutation relations (31) read

$\left[a_{h},a^{\dagger}_{h^{\prime}}\right]=\left[b_{h},b^{\dagger}_{h^{\prime}}\right]=\langle
h|h^{\prime}\rangle\;,$ (34)

where $\langle h|h^{\prime}\rangle$ denotes the Lorentz invariant scalar
product between the test functions $h$ and $h^{\prime}$. i.e.

$\langle
h|h^{\prime}\rangle=\int\frac{d^{3}{\vec{k}}}{(2\pi)^{3}}\frac{1}{2\omega_{k}}{\hat{h}}(\omega_{k},{\vec{k}}){{\hat{h}}}^{{}^{\prime}*}(\omega_{k},{\vec{k}})=\int\frac{d^{4}{k}}{(2\pi)^{4}}2\pi\;\theta(k^{0})\delta(k^{2}-m^{2}){\hat{h}}(k){\hat{h}}^{{}^{\prime}*}(k)\;.$
(35)

In particular, from eq.(34), it follows that

$\left[a_{f},a^{\dagger}_{f}\right]=\left[b_{f},b^{\dagger}_{f}\right]=||f||^{2}\;,$
(36)

where $||f||^{2}$ is the norm of the test function $f$, i.e.

$||f||^{2}=\int\frac{d^{4}{k}}{(2\pi)^{4}}2\pi\;\theta(k^{0})\delta(k^{2}-m^{2}){\hat{f}}(k){\hat{f}}^{*}(k)\;.$
(37)

Let us remark that, by a simple rescaling, the test functions can be taken to
be normalized to 1, namely

$f(x)\rightarrow\frac{1}{||f||}f(x)\;,\qquad||f||=1\;.$ (38)

Therefore, expression (36) becomes

$\left[a_{f},a^{\dagger}_{f}\right]=\left[b_{f},b^{\dagger}_{f}\right]=||f||^{2}=1\;.$
(39)

The vacuum state $|0\rangle$ of the theory is defined as

$a_{f}|0\rangle=b_{g}|0\rangle=0\qquad\forall\;(f,g)\in{\cal
C}_{0}^{\infty}(\mathbb{R}^{4})\;.$ (40)

The scalar complex KG field $\varphi$, eq.(29), enables us to introduce a
squeezed state very similar to that introduced in the previous section,

$|\sigma\rangle=\sqrt{(1-\sigma^{2})}\;e^{\sigma
a^{\dagger}_{f}b^{\dagger}_{g}}|0\rangle\;,\qquad\langle\sigma|\sigma\rangle=1\;,\qquad
0<\sigma<1\;.$ (41)

We write thus

$\displaystyle|\sigma\rangle$ $\displaystyle=$
$\displaystyle\sqrt{(1-\sigma^{2})}\;\sum_{n=0}^{\infty}\sigma^{n}\;\frac{(a_{f}^{\dagger}b_{g}^{\dagger})^{n}}{n!}|0\rangle$
(42) $\displaystyle=$
$\displaystyle\sqrt{(1-\sigma^{2})}\;\sum_{n=0}^{\infty}\left(\sigma^{(2n)}|(2n_{f})(2n_{g})\rangle+\sigma^{(2n+1)}|(2n_{f}+1)(2n_{g}+1)\rangle\right)\;,$

where $|m_{f}m_{g}\rangle$ stands for the normalized state

$|m_{f}m_{g}\rangle=\frac{(a_{f}^{\dagger})^{m}(b_{g}^{\dagger})^{m}}{m!}|0\rangle\;.$
(43)

Proceeding as in the previous example, the operators $A_{i}$, $B_{i}$,
$i=1,2$, are given by

$\displaystyle A_{i}|(2n_{f})\;m_{g}\rangle$ $\displaystyle=$ $\displaystyle
e^{i\alpha_{i}}|(2n_{f}+1)\;m_{g}\rangle\;,\qquad
A_{i}|(2n_{f}+1)\;m_{g}\rangle=e^{-i\alpha_{i}}|(2n_{f})\;m_{g}\rangle\;,$
$\displaystyle B_{i}|m_{f}\;(2n_{g})\rangle$ $\displaystyle=$ $\displaystyle
e^{i\beta_{i}}|m_{f}\;(2n_{g}+1)\rangle\;,\qquad
B_{i}|m_{f}\;(2n_{g}+1)\rangle=e^{-i\beta_{i}}|m_{f}\;(2n_{g})\rangle\;.$ (44)

Again, the operator $A_{i}$ acts only on the first entry, while $B_{i}$ only
on the second one.

It turns out that

$\langle\sigma|\;A_{k}B_{i}\;|\sigma\rangle=\frac{2\sigma}{1+\sigma^{2}}\;\cos(\alpha_{k}+\beta_{i})\;,$
(45)

so that, for the Bell-CHSH correlator, one has precisely the expression
obtained in the case of the squeezed oscillator, namely

$\displaystyle\langle\sigma|{\cal C}_{CHSH}|\sigma\rangle$ $\displaystyle=$
$\displaystyle\langle\sigma|(A_{1}+A_{2})B_{1}+(A_{1}-A_{2})B_{2}|\sigma\rangle$
(46) $\displaystyle=$
$\displaystyle\frac{2\sigma}{1+\sigma^{2}}\left(\cos(\alpha_{1}+\beta_{1})+\cos(\alpha_{2}+\beta_{1})+\cos(\alpha_{1}+\beta_{2})-\cos(\alpha_{2}+\beta_{2})\right)\;.$

Therefore, using

$\alpha_{1}=0\;,\qquad\beta_{1}=-\frac{\pi}{4}\;,\qquad\alpha_{2}=\frac{\pi}{2}\;,\qquad\beta_{2}=\frac{\pi}{4}\;,$
(47)

expression (46) becomes

$\langle\sigma|{\cal
C}_{CHSH}|\sigma\rangle=2\;\frac{2\sqrt{2}\sigma}{1+\sigma^{2}}\;,$ (48)

implying in a violation of the Bell-CHSH inequality for

$\sqrt{2}-1<\sigma<1\;.$ (49)

The maximum violation is attained for values of $\sigma\approx 1$:

$\langle\eta|{\cal C}_{CHSH}|\eta\rangle\approx 2\sqrt{2}\;.$ (50)

It is worth reminding here that the violation of the Bell-CHSH in free Quantum
Field Theory has been established several decades ago by using Algebraic
Quantum Field Theory techniques, see [5] and refs.therein. See also [7] for a
more recent discussion.

##  5 Some remarks on the Minkowski vacuum and the Rindler wedges

Let us conculde with a few remarks on the origin of the violation of the Bell-
CHSH inequality in relativistic Quantum Field Theory. We consider here the
case of a real scalar KG field. It can be argued that the violation of the
Bell-CHSH inequality can be understood in a simple way as a consequence of the
entanglement properties of the vacuum state $|0\rangle$, which can be
expressed as a squeezed state in terms of left and right Rindler modes [8, 9],
i.e.

$|0\rangle=\prod_{i}\left((1-e^{-\frac{2\pi\omega_{i}}{a}})^{\frac{1}{2}}\sum_{n_{i}=0}^{\infty}\;e^{-\frac{\pi
n_{i}\omega_{i}}{a}}\;|n_{i}\rangle_{L}|n_{i}\rangle_{R}\right)\;,$ (51)

where $(|n_{i}\rangle_{L},|n_{i}\rangle_{R})$ are the left and right Rindler
modes and $T=\frac{a}{2\pi}$ is the Unruh temperature. The relation (51)
follows from the use of a Bogoliubov transformation applied to the the
quantization of the real scalar KG field in the Rindler wedges [8, 9].
Proceeding as in the previous cases, for the four operators
$(A_{k},B_{k}),k=1,2$, we have

$\displaystyle A_{k}|2n_{i}\rangle_{L}$ $\displaystyle=$ $\displaystyle
e^{i\alpha_{k}}|2n_{i}+1\rangle_{L}\;,\qquad
A_{k}|2n_{i}+1\rangle_{L}=e^{-i\alpha_{k}}|2n_{i}\rangle_{L}\;,$
$\displaystyle B_{k}|2n_{i}\rangle_{R}$ $\displaystyle=$ $\displaystyle
e^{i\beta_{k}}|2n_{i}+1\rangle_{R}\;,\qquad
B_{k}|2n_{i}+1\rangle_{R}=e^{-i\beta_{k}}|2n_{i}\rangle_{R}$ (52)
$\displaystyle;.$ (53)

As a consequence, setting again

$\alpha_{1}=0\;,\qquad\alpha_{2}=\frac{\pi}{2}\;,\qquad\beta_{1}=-\frac{\pi}{4}\;,\qquad\beta_{2}=\frac{\pi}{4}\;,$
(54)

the Bell-CHSH inequality can be parametrized as

$|\langle\Omega|{\cal C}_{CHSH}|\Omega\rangle|=2{\sqrt{2}}\;\tau(T)\;,$ (55)

where the form factor $\tau$ reads

$\tau(T)=2\sum_{i}\frac{\left(e^{\frac{\pi\omega_{i}}{a}}-e^{-\frac{\pi\omega_{i}}{a}}\right)}{\left(e^{\frac{2\pi\omega_{i}}{a}}-e^{-\frac{2\pi\omega_{i}}{a}}\right)}=\sum_{i}\frac{1}{\cosh(\frac{\omega_{i}}{2T})}\;,$
(56)

from which it follows that the violation of the Bell-CHSH inequality in
relativistic Quantum Field Theory can be anlyzed in terms of the Unruh
temperature [10].

##  6 Conclusion

In this work we have pointed out that the four operators $(A_{i},B_{i}),i=1,2$
entering the Bell-CHSH inequality can be constructed in a simple and elegant
way. The setup turns out to be of general applicability, ranging from Quantum
Mechanics examples to relativistic Quantum Field Theory.

We have argued that the violation of the Bell-CHSH inequality in relativistic
Quantum Field Theory can be understood as a direct consequence of the
decomposition of the vacuum sate $|0\rangle$ as a deeply entangled squeezed
state in terms of left and right Rindler modes.

These considerations might result in applications in several Quantum Field
Theory models, including Abelian and non-Abelian gauge theories [10].

## Acknowledgements

The authors would like to thank the Brazilian agencies CNPq and FAPERJ for
financial support. S.P. Sorella is a level $1$ CNPq researcher under the
contract 301030/2019-7.

## References

  * [1] J. S. Bell, Physics Physique Fizika 1, 195-200 (1964) doi:10.1103/PhysicsPhysiqueFizika.1.195
  * [2] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23, 880-884 (1969) doi:10.1103/PhysRevLett.23.880
  * [3] B .S . Cirelson, Lett. Math. Phys. 4, 93-100, (1980)
  * [4] J . C. Howell, A . Lamas-Linares, D . Bouwmeester, Phys. Rev. Lett. 88, 030401 (2002)
  * [5] Stephen J. Summers and Reinhard Werner, J. Math. Phys. 28, 2448 (1987); doi: 10.1063/1.527734
  * [6] R. Haag, Local quantum physics: Fields, particles, algebras, Springer-Verlag, 1992
  * [7] G. Peruzzo and S. P. Sorella, Phys. Rev. D 106, no.12, 125020 (2022) doi:10.1103/PhysRevD.106.125020 [arXiv:2210.16372 [hep-th]].
  * [8] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, Rev. Mod. Phys. 80, 787-838 (2008) doi:10.1103/RevModPhys.80.787 [arXiv:0710.5373 [gr-qc]].
  * [9] D. Harlow, Rev. Mod. Phys. 88, 015002 (2016) doi:10.1103/RevModPhys.88.015002 [arXiv:1409.1231 [hep-th]].
  * [10] In preparation


# ALMA and Keck analysis of Fomalhaut field sources: JWST’s Great Dust Cloud
is a background object

Grant M. Kennedy1, Joshua B. Lovell2, Paul Kalas3,4,5 & Michael P. Fitzgerald6
1Department of Physics and Centre for Exoplanets and Habitability, University
of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
2Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge,
MA 02138-1516, USA
3 University of California, Berkeley, 501 Campbell Hall, 3411 Berkeley, CA
94720-3411, USA
4Institute of Astrophysics, FORTH, GR-71110 Heraklion, Greece
5SETI Institute, Carl Sagan Center, 189 Bernardo Ave., Mountain View, CA
94043, USA
6Department of Physics and Astronomy, University of California, Los Angeles,
430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547, USA E-mail:
<EMAIL_ADDRESS>
(Accepted 2023 July 4. Received 2023 July 4; in original form 2023 May 17)

###### Abstract

At 7.7 pc, the A-type star Fomalhaut hosts a bright debris disk with multiple
radial components. The disk is eccentric and misaligned, strongly suggesting
that it is sculpted by interaction with one or more planets. Compact sources
are now being detected with JWST, suggesting that new planet detections may be
imminent. However, to confirm such sources as companions, common proper motion
with the star must be established, as with unprecedented sensitivity comes a
high probability that planet candidates are actually background objects. Here,
ALMA and Keck observations of Fomalhaut are found to show significant emission
at the same sky location as multiple compact sources in JWST MIRI
coronagraphic observations, one of which has been dubbed the "Great Dust
Cloud" because it lies within the outer belt. Since the ground-based data were
obtained between 6 to 18 years prior to the JWST observations, these compact
sources are unlikely to be common proper motion companions to Fomalhaut. More
generally, this work illustrates that images collected at a range of
wavelengths can be valuable for rejecting planet candidates uncovered via
direct imaging with JWST.

###### keywords:

stars: individual: Fomalhaut – infrared: planetary systems – submillimetre:
planetary systems

††pubyear: 2023††pagerange: ALMA and Keck analysis of Fomalhaut field sources:
JWST’s Great Dust Cloud is a background object–ALMA and Keck analysis of
Fomalhaut field sources: JWST’s Great Dust Cloud is a background object

##  1 Introduction

Fomalhaut is a bright ($m_{V}$=1.16 mag) nearby (7.7 pc) A-type star known to
host a bright circumstellar dust disk (Gillett, 1986). The system has been
studied in detail for decades, yielding optical images of dust-scattered light
and a compact dusty source (Kalas et al., 2005, 2008), and far-infrared and
mm-wave images of dust thermal emission (e.g. Holland et al., 2003;
Stapelfeldt et al., 2004; MacGregor et al., 2017) and gas (Matrà et al.,
2017).

A key question for Fomalhaut, as with all stellar systems, is “where are the
planets?” While in the past such a question might have seemed to be
optimistic, the _Kepler_ mission (Borucki et al., 2003) has proven beyond
doubt that planets are common. In addition, Fomalhaut’s bright debris ring is
eccentric (Kalas et al., 2005), for which the simplest explanation is that the
ring material is perturbed by an eccentric planet (Wyatt et al., 1999). Thus,
this system has been, and will continue to be, a prime target in the ongoing
search for planets that can be directly imaged.

The possibility of planets has also been highlighted by new images presented
by Gáspár et al. (2023) from the Mid-Infrared Instrument (MIRI; Wright et al.,
2008) on the James Webb Space Telescope (JWST; Gardner et al., 2006). These
data show that the Fomalhaut disk has additional structure that has not been
previously resolved. While an asteroid belt analog (ABA) at $\sim$11 au has
long been known to exist (e.g. Stapelfeldt et al., 2004; Acke et al., 2012; Su
et al., 2013) and is confirmed with JWST, the images reveal a previously
unseen intermediate belt at $\sim$90 au that may be misaligned with the outer
belt. The outer belt at $\sim$140 au is dubbed the Kuiper Belt Analog (KBA) by
Gáspár et al. (2023). With a series of radially nested belts that are
separated by gaps, Fomalhaut somewhat resembles an eccentric and misaligned
version of the ringed protoplanetary disks (e.g. ALMA Partnership et al.,
2015; Andrews et al., 2018). These continuum radial structures in
protoplanetary disks are thought to be caused by a combination of planetary
perturbation and dust drift (e.g. Rice et al., 2006; Pinilla et al., 2012), so
it seems reasonable that systems such as Fomalhaut inherit a similar structure
of planets and dusty rings.

Another JWST discovery at 23.0 and 25.5 $\mu$m is a compact structure that
lies west of the star within the KBA, dubbed the “Great Dust Cloud” (GDC).
This source is 89 $\mu$Jy at 25.5 $\mu$m, compared to the star’s 2.5 Jy, so
extremely faint. Because it appears in both the 23.0 and 25.5 $\mu$m data, it
is confirmed as a real astrophysical source as opposed to an instrumental or
data reduction artifact. The fact that the GDC lies almost perfectly within
Fomalhaut’s narrow KBA (i.e. a special location in the image) suggests that it
is less likely to be a background source, though Gáspár et al. make clear that
with only a single epoch of JWST observation the possibility that the GDC is a
background source cannot be ruled out. The idea that dust clumps could be
created and reside within a debris disk is not new; Wyatt & Dent (2002)
developed a theory for collisional clumps, and in the case of the Fomalhaut
disk, suggested that these may be detectable with ALMA and/or a future large-
aperture mid-infrared space telescope (i.e. JWST).

In this work we aim to test whether the putative dust cloud is indeed
associated with Fomalhaut; the system’s proper motion of $\mu_{\alpha}=329.0$
mas yr-1 and $\mu_{\delta}=-164.7$ mas yr-1 (van Leeuwen, 2007) means that
previously detected background sources may now lie behind the KBA,
masquerading as dust clouds. We revisit archival ALMA data for this system
from 2015–2016, which show a source that lies at the sky location of the GDC.
We validate our finding that the GDC is a background object with previously
unpublished Keck adaptive optics (AO) data of Fomalhaut obtained at multiple
epochs obtained 11–18 years before the JWST observations were made.

##  2 Data

The ALMA 1.3 mm continuum data have appeared in two publications (MacGregor et
al., 2017; Kennedy, 2020), and the processed JWST data (GTO-1193; PI Beichman)
have been made available in FITS format by Gáspár et al.
(2023).111https://github.com/merope82/Fomalhaut Readers can refer to those
publications for details on their reduction and calibration procedures, but we
will review the most relevant points here. A portion of the Keck data were
used in Kalas et al. (2008) where the objective was to detect Fomalhaut b
rather than make a map of background sources. In the present analysis we
include subsequent epochs of Keck $H$-band observations made through 2011
(Table LABEL:sec:keckobs).

###  2.1 ALMA

Atacama Large Millimetre/Submillimetre Array (ALMA) observations of the entire
Fomalhaut ring were obtained in late 2015 December and early 2016 January in
Band 6 (1.3 mm) (MacGregor et al., 2017). They are a mosaic of seven pointings
that can be combined to make a single large image. Being an interferometer,
ALMA’s sky sensitivity is dictated by its primary beam, with a full-width at
half-maximum of approximately 22 ″; while the sensitivity in uncorrected
images appears uniform, the signal is attenuated by the primary beam and is in
fact significantly noisier farther from the image center. Thus, to be
detected, sources farther from the image center must be brighter. The spatial
resolution of ALMA is set by relative antennae positioning and spacing at the
time of observation, which for this data provide a beam with 1.6${\times}$1.2
″ resolution. The clean images show that the star is strongly detected and
lies within a few tens of mas of the expected coordinates at the observation
epoch.

These data were also presented in Kennedy (2020) who analyzed the KBA width by
creating a model fit to the observations (the ABA and intermediate belt were
not detected). To demonstrate the fidelity of the model, their Fig. 6 shows
the difference image after Fomalhaut’s dust belt is subtracted by the model.
Black contours identified residual emission in the entire field greater than
3$\sigma$ above the noise, where $\sigma=13$ $\mu$Jy. In the present analysis
we will discuss residual emission sources 4$\sigma$ above the noise.

###  2.2 Keck

Table 1: Keck $H$-band observing log. UT | MJD | Nexp | tint |  $\Delta$P.A.
---|---|---|---|---
| (midpoint) |  | (s) | (deg)
2005-07-17 | 53568.59 | 117 |  03790.0 | 41.1
2005-07-27 | 53578.54 | 144 |  04320.0 | 38.4
2005-07-28 | 53579.55 | 163 |  04890.0 | 44.8
2005-10-21 | 53664.31 | 174 |  05197.3 | 44.8
2010-07-02 | 55379.55 | 359 | 10770.0 | 69.2
2010-07-03 | 55380.55 | 404 | 12105.0 | 71.8
2010-11-11 | 55511.26 | 217 |  05610.0 | 41.1
2011-07-12 | 55754.57 | 138 |  02028.6 | 27.5
2011-07-14 | 55756.55 | 215 |  09178.0 | 66.2
Table 2: IDs and positions of sources (J2000).

ID | RA | Dec | JWST | Keck | ALMA
---|---|---|---|---|---
K01 | 22:57:37.44 | -29:37:13.86 |  | y |
K02 | 22:57:37.57 | -29:37:22.51 |  | y |
K03 | 22:57:37.61 | -29:37:31.21 |  | y |
K04 | 22:57:38.08 | -29:37:32.25 | y | y |
K05 | 22:57:38.16 | -29:37:17.86 |  | y |
K06 | 22:57:38.24 | -29:37:12.97 |  | y |
A01 | 22:57:38.31 | -29:37:05.52 |  |  | y
K07 | 22:57:38.79 | -29:37:34.25 |  | y |
D01* | 22:57:38.88 | -29:37:04.84 |  |  | y
K08 | 22:57:39.09 | -29:37:44.15 | y | y |
A02* | 22:57:39.15 | -29:37:25.59 | y | y | y
K09* | 22:57:39.15 | -29:37:26.18 | y | y | y
GDC-25* | 22:57:39.16 | -29:37:26.22 | y | y | y
GDC-23* | 22:57:39.18 | -29:37:26.18 | y | y | y
K10 | 22:57:39.34 | -29:37:04.03 |  | y |
K11* | 22:57:39.34 | -29:37:05.20 |  | y |
K12* | 22:57:39.80 | -29:37:28.79 | y | y |
K13* | 22:57:40.07 | -29:37:36.81 |  | y |
D02* | 22:57:40.15 | -29:37:35.12 |  |  | y
K14* | 22:57:40.17 | -29:37:29.93 | y | y |
K15* | 22:57:40.41 | -29:37:34.53 | y | y |
K16 | 22:57:40.62 | -29:37:34.19 |  | y |
A03, K17 | 22:57:40.74 | -29:37:32.43 | y | y | y
K18 | 22:57:41.02 | -29:37:08.22 |  | y |
K19 | 22:57:41.04 | -29:37:25.58 | y | y |
A04 | 22:57:41.45 | -29:37:21.54 | y |  | y

Notes: The last three columns indicate which instrument sources were detected
by (JWST refers to either 23 or 25.5 $\mu$m). Starred sources lie on or
interior to the KBA. The IDs A02, K09, GDC-23 (JWST 23.0 $\mu$m), and GDC-25
(JWST 25.5 $\mu$m) are all the same source. Likewise A03 and K17 are the same
source.

At Keck II, Fomalhaut was imaged with AO and NIRC2 (Wizinowich et al., 2000)
using its widest field of 40$\times$40 ″ (40 mas pix-1) at 1.6 $\mu$m. The
star is sufficiently bright to serve as its own wavefront reference with a 1
kHz readout rate for the wavefront sensor, along with a neutral density filter
in the wavefront sensing optical path. As a southern target, we generally
attempted to observe it from the time it reached a reasonable elevation angle
until the telescope could no longer track it due to the Nasmyth deck pointing
limit. The image rotator located at the Nasmyth focus was placed in vertical
angle mode, which stabilizes the orientation of the telescope pupil relative
to the detector, and enables angular differential imaging (ADI). Due to the
brightness of Fomalhaut ($m_{R}$=1.11 mag, $m_{H}$=1.05 mag), we used a large
(2.0 ″ diameter) focal plane mask in order to prevent electronic artifacts in
detector readout. We chose exposure times to allow for deep integration,
resulting in saturation and nonlinearity in a region extending radially beyond
the edge of the focal plane mask. The size of this region varied with
conditions and the degree of AO correction. With good conditions and AO
performance, the PSF full-width at half-maximum is $\sim$60 mas (undersampled
relative to the Nyquist limit).

Observations were classically scheduled and weather conditions varied. Cloud
cover and the degree of atmospheric turbulence had significant impacts on the
sensitivity achieved at any given epoch. Individual exposures were reduced
using standard techniques to correct for electronic biases, flat field (with
the focal plane mask in place), detector nonlinearity, and field distortion.

To subtract Fomalhaut’s point spread function (PSF) we used the LOCI technique
on these ADI data (Lafrenière et al., 2007). For a given exposure, this method
combines other images in the sequence to construct a reference PSF. The data
for each night were processed separately.

The data obtained throughout each night and from night-to-night are subject to
variable extinction. The PSF of all sources will also change over time with
variations in the on-axis AO correction that is sensitive to changes in the
vertical profiles of atmospheric turbulence strengths and wind velocities,
quasi-static aberrations, etc. In addition, the time variability of high-
altitude atmospheric turbulence introduces fluctuations in the structure of
the off-axis AO PSF.

Figure 1: Fomalhaut’s belt features after high-pass filtering the 23.0 (left
panel) and 25.5 $\mu$m data (right panel). Both images have the same size and
center. Arrows show the stellar proper motion since 2005 July (left panel) and
2015 December (right panel). Keck sources and A04 are circled and labelled.
The circles are approximately 200 mas in diameter. The 23.0 $\mu$m image in
the left panel has been moved 0.3 ″ W to account for a difference in the
apparent GDC location between 25.5 and 23.0 $\mu$m.

Since it is detectable in individual exposures, we use the time variability of
the bright star southeast of Fomalhaut (K13, see Table 2) to track changes in
the AO performance and extinction. This allows us to combine images during
each night and across nights by assigning a weight factor that incorporates
the signal recovered from the background star. We also incorporate a measure
of the PSF-subtraction fidelity, by using an estimate of the subtraction
uncertainty derived from the $\chi^{2}$ measure of the LOCI optimization. We
use a robust combination method that incorporates the above weights. All
images across all nights of observations in a given band are combined after
registration to the celestial reference frame, thus co-adding the signal from
background sources.

Any sources that have a common proper motion with Fomalhaut would appear as
triplet images in the combined frame corresponding to the 2005, 2010, and 2011
epochs of observation (there are no such sources detected). All other sources
are background objects which we can further verify by blinking the data from
the three epochs and using additional versions of the images that are binned
and smoothed. When these data from different epochs are registered to the
position of Fomalhaut, blinking shows the background sources moving along an
axis oriented at PA = 116.59$\degr$ (Fomalhaut’s proper motion vector)
relative to any local noise. Thus, even though source K09 lies close to the
central speckle halo, we confirmed it as a real astrophysical object using
this method eight years before JWST was launched. Therefore K09 was identified
independently from the JWST results, so as with the ALMA sources there is no
confirmation bias.

In Table 2 we provide the centroid positions of background sources detected
and validated by our Keck program. We adopt a conservative position
uncertainty estimate of 80 mas to account for the systematic and random errors
involved in determining the position of Fomalhaut, the true north orientation,
uncorrected geometric distortions, and the centroiding measurement itself that
depends on the signal-to-noise ratio (SNR) and morphology of each source.
Several of our sources have been previously noted as background objects by
Currie et al. (2013); their source 1 is K13, 2=K04, 3=K07, 4=K10, 5=K11,
6=K14, and 7=K12. With the exception of K09 below, we have not presented
magnitudes as many sources are resolved with a size of a few tens of mas and
hence may be affected by the self-subtraction that is inherent in LOCI images.
As an approximate reference, K13 is Gaia DR3 6606098685561921280 with a $G$
magnitude of $18.22\pm 0.02$ and a $G_{BP}-G_{RP}$ colour consistent with
zero, suggesting that the $H$-band magnitude is similar.

###  2.3 JWST

JWST’s MIRI field of view (FOV) at 25.5 $\mu$m is large: 56$\times$56 ″ in
these Fomalhaut observations using the “BRIGHTSKY” subarray. MIRI can capture
the entire Fomalhaut disk in a single FOV, retaining high sensitivity to the
dusty outer regions and to compact background sources that may be
significantly separated from Fomalhaut’s disk, as well as any that could be
coincident with it. The 23.0 $\mu$m data have a smaller FOV (30$\times$30 ″)
that do not capture the southeast and northwest portions of Fomalhaut’s KBA.
The angular resolution of MIRI is higher than the ALMA observations,
approximately 0.5 ″. We do not carry out a complete search for point sources
in the MIRI fields, but identify those that are also detected with ALMA and/or
Keck. The GDC is not visible at 15.5 $\mu$m, so we do not consider those data
here.

To assist in measuring the positions of features in the KBA and intermediate
belt, we high-pass filtered the MIRI data. We smoothed each image with a
Gaussian ($\sigma$=2 pixels) and subtracted the smoothed images from the
originals. The difference images are smoothed by a Gaussian ($\sigma$=1 pixel
at 23.0 $\mu$m, $\sigma$=3 pixels at 25.5 $\mu$m) and shown in Figure 1, with
our measurements of GDC positions listed as GDC-23 and GDC-25 in Table 2.

Our centroid measurements for GDC reveal that the 23.0 $\mu$m position lies
0.3 ″ east of its 25.5 $\mu$m position. Blinking the two JWST images when
registered to the world coordinate system shows that the intermediate belt and
KBA also have the same shift between images. This shift is evidence for a
systematic error in determining the location of Fomalhaut on the detector at
each wavelength or some other factor in the data processing that establishes
the celestial reference frame. Our analysis requires a comparison of source
positions in the time domain at different wavelengths. However, since this is
a relatively sparse field, source confusion is not a significant problem and
the 0.3 ″ position uncertainty for JWST astrometry is unlikely to change our
conclusions. The GDC position could serve as a guide since Table 2 shows that
the 25.5 $\mu$m position instead of the 23.0 $\mu$m position agrees more
closely with both the ALMA and Keck data.

The JWST observations were carried out on 2022 October 22, nearly seven years
after the ALMA observations and more than 11 years after the last Keck
observation. Given its southeastward proper motion of 367.9 mas/yr, Fomalhaut
at the JWST epoch has moved 2.6 ″ away from the ALMA epoch and 4.1 ″ away from
the last Keck epoch. In contrast, background sources typically have negligible
proper motions, and so would remain static on the sky plane relative to
Fomalhaut. Given these high angular resolution observations with Keck, ALMA
and JWST, the background sources are easily distinguishable between
observations.

##  3 Results

We now compare the ALMA and Keck data to JWST, focusing on the locations of
compact sources.

###  3.1 Keck

Figure 2: 2011 Keck $H$-band image with a log stretch and background sources
circled and labelled. The arrow shows the proper motion from 2011 July to 2022
October. The circles in the main panel are approximately 150 mas in diameter.
The insets show 4$\times$4 ″ zooms centered on the GDC and K14 which is
extended along PA$\sim$30$\degr$.

The background sources discovered in the multi-epoch Keck data are labeled in
Figure 2 and overlayed on the JWST 23.0 and 25.5 $\mu$m images in Figure 1.
Source K09 is coincident with the GDC and detected at SNR=5. At our epochs of
observation it appears as a point source at $\sim$5 ″ projected radius from
Fomalhaut. It is possible that K09 is extended but the residual noise at that
radius after subtracting Fomalhaut’s bright PSF and the effect of self-
subtraction from the LOCI algorithm interferes with the detection of diffuse
extended structure. Using a Keck/NIRC2 $H$-band zeropoint of 24.2 mag, K09 is
20.9$\pm$0.3 mag. However, this estimate is probably fainter than the true
magnitude due to self-subtraction from ADI/LOCI processing. K09 is offset 0.59
″ S from the ALMA source A02 discussed below but aligned in the right
ascension direction. K09 is 0.12 ″ W and 0.05 ″ N of the 25.5 $\mu$m position
(GDC-25, Table 2). The discrepancy in right ascension is larger between K09
and GDC-23 (see discussion in Section 2.3) but negligible in declination.

Table 2 lists seven more background sources in the Keck data that appear to
have counterparts in the JWST data. K04 is a background galaxy because it
appears extended in both the Keck data analyzed here as well as the HST/STIS
optical data published in Kalas et al. (2013). K08 has a counterpart in the
JWST data that is more easily distinguished by binning and smoothing the 25.5
$\mu$m data instead of high-pass filtering. It has an extended morphology
consistent with a background galaxy. The same is true for K17, which is
located outside the Keck FOV in several nights of data, but within the FOV in
other observations (2010-07-02, 2010-07-03, and 2011-07-14). K17 is co-
incident with A03 which was also discussed in MacGregor et al. (2017) who show
an HST/STIS image of the source. K19 corresponds to emission on the eastern
edge of the 23.0$\mu$m frame (Figure 1) that is also apparent in the 25.5
$\mu$m image using binning and smoothing instead of high-pass filtering. It
also has an elliptical morphology in the Keck data consistent with a galaxy.

Figure 3: ALMA images. The left panel shows the continuum image with 4
$\sigma$ residuals as contours, and the right panel shows the residuals with a
10 $\sigma$ contour from the continuum map. The arrows show the proper motion
from 2015 December to 2022 October. The inset shows a 6$\times$6 ″ zoom of the
residual image, showing the sources A02 and K09 (noting that GDC is at almost
exactly the same location as K09). The beam is shown in the lower left corner
of each image. The image is not primary beam corrected, so the flux scale is
not uniform across the image. Figures and inset have the same sizes and
centers as Fig. 4.

Three more Keck sources are superimposed on or inside of Fomalhaut’s KBA and
may influence the interpretations of belt morphologies from JWST data. K12 may
make the ABA appear brighter and more extended to the southeast of the star.
K14 may be misinterpreted as a clump or spiral feature along the intermediate
belt, possibly influencing measurements of the belt edges or brightest points
needed to derive its position angle and inclination (see also the inset for
Fig. 2 that shows this source is resolved with an elliptical morphology). K15
is coincident with the KBA and may lead to the same misinterpretation as the
GDC. Finally, we note that even though K11 overlaps with the northern part of
the belt, it does not appear to have a counterpart in 25.5 $\mu$m emission.

###  3.2 ALMA

Figure 4: ALMA - JWST comparison. The colour map is the JWST image, the
contours are the ALMA residuals, and the labelled sources are significant ALMA
residuals. The arrow shows the proper motion from 2015 December to 2022
October. The insets show 6$\times$6 ″ zooms near A04 and the GDC. The figure
and insets have the same sizes and centers as Fig. 3.

The ALMA data, both the image and residuals after model subtraction, are shown
in Figure 3 (reproduced from Kennedy, 2020). The sources identified with
emission above 4$\sigma$ (where $\sigma=13$ $\mu$Jy) are given in Table 2,
none of which are well resolved. The residual labelled A02 only becomes
clearly visible once a model is subtracted. For clarity the primary beam
correction has not been applied, so the flux scale is not constant across the
image. A minor difference relative to the images shown in Kennedy (2020) is
that the field of view has been increased to incorporate the source E of the
star labelled A04. With a beam slightly larger than 1 ″, and low SNR
detections, we conservatively estimate the positional uncertainties to be
approximtely 1 ″.

A number of other significant contours are present in the residuals. Two of
these lie near the disk ansae (D01, D02), and are thought to be associated
with imperfect modelling of the disk surface brightness, though this
conclusion is more secure for D01 as D02 is not exactly at an ansa. As shown
by the right panel of Figure 3, the model is otherwise a very good
representation of the data, so there is no reason to believe that the residual
near the GDC is an artefact of the modelling. Aside from these residuals the
other three significant residuals lie well away from the disk. While A04
appears faint, at this location the primary beam power is about 20% relative
to the image center, so this source is actually about a factor of two brighter
than A03. By looking at individual scans we see no evidence that any of the
ALMA sources are variable.

As with Keck, there is a clear correspondence between some of the ALMA and
JWST sources. Figure 4 combines the ALMA contours and JWST 25.5 $\mu$m image
in a single figure. It is clear that A02, A03, and A04 correspond to JWST
sources, and that these sources are therefore almost certainly background
objects that are detected at both wavelengths (with A02 and A04 also detected
by Keck). In terms of fluxes this conclusion also seems reasonable; these
sources have fluxes that are similar within a factor of a few for both ALMA
and JWST, so if they are similar types of galaxies, then their detections in
both datasets is reasonably expected. The source A01 is not obviously detected
with JWST, but may be detected with deeper imaging, or perhaps at other JWST
wavelengths (i.e. with NIRCAM).

As shown by the insets in Figure 4, for A02 and A04 there are small ($<$0.6 ″)
offsets to the S of the JWST sources relative to the ALMA contours, suggesting
that there is a slight offset in coordinate systems. This offset cannot be
discerned for A03, but this source is probably marginally resolved by ALMA and
detected at low SNR with JWST, so any offset is harder to distinguish. The
offset is slightly greater for A02, but the residuals in Figure 3 show that
the location of this source may be biased north by a negative residual to the
south.

In summary, as with Keck, the main conclusion from comparing the ALMA and JWST
data is that the GDC is a background object.

###  3.3 The Great Dust Cloud: star or galaxy?

In terms of the Fomalhaut system, whether the GDC is a star or galaxy in the
background is not important. But JWST will no doubt face similar issues in the
future, so we briefly consider the possibilities.

First considering the nature of the GDC, a stellar source is unlikely because
it would require a bright infrared excess component; the Keck flux is about 4
$\mu$Jy and therefore roughly 10–20 times fainter than in the mid-IR and
millimeter fluxes, which implies $L_{\rm excess}/L_{\star}~{}>~{}10$%, a level
seen for bright protoplanetary disks. The spectrum is rising from 23 to 25.5
$\mu$m (Gáspár et al., 2023), which implies the peak is somewhere in the far-
IR at a level higher than seen with JWST, which could easily imply $L_{\rm
excess}/L_{\star}\gtrsim 1$, which is energetically impossible unless the
excess component is self-luminous (which is not normally the case for
circumstellar material). Such a spectrum is however consistent with an
ultraluminous infrared galaxy (ULIRG, e.g. Armus et al., 2020). Thus, based on
the spectrum alone, we conclude that it is most likely that the GDC is a
background galaxy. We cannot however be definitive here because if the Keck
source is extended, the flux could be higher, and thus the energetic argument
weaker.

What is the likelihood that a background galaxy could be located directly
behind the Fomalhaut debris disk, thus masquerading as a dust cloud or planet
candidate? This question has been asked previously with ALMA observations of
debris disks, albeit for much lower sensitivity data (e.g. Lovell et al.,
2021, 2022). Following a similar methodology, using the 1.2 mm sub-millimeter
galaxy (SMG) counts of González-López et al. (2020), we calculate the number
of SMGs on the sky with a flux ${>}50\,\mu$Jy (i.e. $\approx$4$\sigma$) as
approximately 37,000. Given the angular size of the Fomalhaut debris disk,
i.e. an annulus with a radius $r=136$ au and width $\Delta r=27$ au inclined
at 66∘, at its distance of 7.7 pc we estimate that $0.5\pm 0.05$ SMGs should
appear behind the disk (accounting for the ALMA noise of $13\,\mu$Jy). While
this same count rate would predict approximately ten sources appearing in
entire ALMA image, it likely detects fewer because the ALMA sensitivity is not
uniform across this region. This estimate suggests there is a fairly high
probability that the ALMA source A02 located just inside the dust ring is a
background galaxy, and the same applies to the exterior sources.

Mid-IR galaxy counts at 21 $\mu$m (with JWST MIRI; Wu et al., 2022), 24 $\mu$m
(with Spitzer; Papovich et al., 2004), and from models at 25.5 $\mu$m (with
JWST MIRI; Cowley et al., 2018) predict similar number counts to ALMA;
respectively 0.1–2, 6–8, and 4–8 for sources above 40$\mu$Jy in a 56$\times$56
″ region on the sky (i.e. the MIRI FOV), and 0.01–0.2, 0.6–0.8, and 0.4–0.8
for the area of the disk annulus (where our threshold choice of 40$\mu$Jy
corresponds to about 3$\times$ the RMS of the MIRI image). Empirically, the
MIRI image suggests that the count is close to ten per FOV, in good agreement
with Cowley et al. (2018), Papovich et al. (2004) and the ALMA counts.
Although the counts are somewhat higher than Wu et al. (2022), the predictions
of this latter study have large reported uncertainties, and their difference
could be due to cosmic variance.

Based on reasonable consistency with the expected number of detections from
galaxy counts, and the total number of sources present in the images, we
consider it probable that most, if not all, of the compact sources seen in the
ALMA and JWST images are background galaxies. In the case of GDC, the flux
levels at optical, mid-IR, and mm wavelengths, also argue in favor of a
background galaxy.

##  4 Conclusions

By considering ALMA and Keck data, and the newly presented JWST MIRI data of
Fomalhaut, we conclude that the source dubbed the “Great Dust Cloud” is a
chance alignment with a background source. Each of the ALMA and Keck datasets
independently detect multiple sources in common with JWST. For both ALMA and
Keck a source is found at the sky location of the GDC, when an object that is
co-moving with Fomalhaut would have moved 2-4 ″ between observations. Galaxy
number counts suggest that at the high sensitivity provided by JWST, a
coincidence such as a source like the GDC appearing within the disk was
probable. The spectrum of the GDC also suggests that it is a background
galaxy.

We also detect a number of other background sources, some of which are co-
located with different Fomalhaut disk components as observed by JWST, and
could influence geometric interpretations of Fomalhaut’s disk. Given
Fomalhaut’s proper motion direction, many of the background sources that are
not currently behind the disk will pass behind it in the future. Most of these
are not apparent in the JWST images in Figure 1 so are unlikely to be a
problem.

We find that there is a small astrometric shift between the 23 and 25.5 $\mu$m
images, with the former being 0.3 ″ to the east. This offset is visible for
the GDC and other disk features.

More generally, we conclude that new direct imaging campaigns with JWST will
be sufficiently sensitive to detect galaxies that are also visible in deep
($\sigma\lesssim 10$ $\mu$Jy) ALMA observations, and that faint objects
detected with JWST may also be detected with optical/near-IR high-contrast
imaging. At least some JWST direct imaging programmes aim to detect planets in
nearby systems that also host well-known disks, so it is possible that
previously obtained ALMA data will help distinguish planets from background
sources in other systems.

## Acknowledgements

We thank the referee for a succinct report. GMK is supported by the Royal
Society as a Royal Society University Research Fellow. JBL acknowledges
support from the Smithsonian Institution as a Submillimeter Array (SMA)
Fellow. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2015.1.00966.S ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. This work is based on observations made with the NASA/ESA/CSA James Webb
Space Telescope. These observations are associated with program 1193. Some of
the data presented herein were obtained at the W. M. Keck Observatory, which
is operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation. The authors wish to recognize
and acknowledge the very significant cultural role and reverence that the
summit of Maunakea has always had within the indigenous Hawaiian community. We
are most fortunate to have the opportunity to conduct observations from this
mountain.

## Data Availability

The ALMA data used here are available in the ALMA archive at
https://almascience.eso.org/aq. A script to split and image the ALMA data
associated with Kennedy (2020) is at https://github.com/drgmk/eccentric-width.
The final JWST images associated with Gáspár et al. (2023) are available at
https://github.com/merope82/Fomalhaut. Keck data are available in the Keck
Observatory Archive at https://www2.keck.hawaii.edu/koa/public/koa.php.

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