# Large sets containing no copies of a given infinite sequence

Mihail N. Kolountzakis  Department of Mathematics and Applied Mathematics,
University of Crete, Voutes Campus, 70013 Heraklion, Crete, Greece.
<EMAIL_ADDRESS>and  Effie Papageorgiou  Department of Mathematics and
Applied Mathematics, University of Crete, Voutes Campus, 70013 Heraklion,
Crete, Greece<EMAIL_ADDRESS>
(Date: August 4, 2022. Revised August 19, 2023.)

###### Abstract.

Suppose $a_{n}$ is a real, nonnegative sequence that does not increase
exponentially. For any $p<1$ we construct a Lebesgue measurable set
$E\subseteq{\mathbb{R}}$ which has measure at least $p$ in any unit interval
and which contains no affine copy ${\left\\{{x+ta_{n}:\
n\in{\mathbb{N}}}\right\\}}$ of the given sequence (for any
$x\in{\mathbb{R}},t>0$). We generalize this to higher dimensions and also for
some “non-linear” copies of the sequence. Our method is probabilistic.

###### Key words and phrases:

Erdős similarity problem, Euclidean Ramsey theory, Probabilistic Method

###### 2020 Mathematics Subject Classification:

28A80, 05D40

Supported by the Hellenic Foundation for Research and Innovation, Project
HFRI-FM17-1733 and by University of Crete Grant 4725

###### Contents

  1. 1 Introduction
  2. 2 Warm-up and some basic tools: no translational copies
  3. 3 No affine copies for slowly increasing sequences
    1. 3.1 Proof of Lemma 3.1
  4. 4 The problem in higher dimension

##  1\. Introduction

In Euclidean Ramsey Theory one is interested in assuming some kind of
largeness for sets $E$ in Euclidean space ${\mathbb{R}}^{d}$, or, sometimes in
${\mathbb{Z}}^{d}$, and concluding that $E$ then contains a “copy” of a
pattern. The most famous such example is perhaps Szemeredi’s Theorem [Sze75]
which states that any subset of the integers with positive density contains
aribtrarily long arithmetic progressions. Another well known example is the
theorem of Falconer and Marstrand [FM86], Furstenberg, Katznelson and Weiss
[FKW90] and Bourgain [Bou86] (see also [Kol04]) that if the set
$E\subseteq{\mathbb{R}}^{d}$ has positive Lebesgue density (this means that
there are arbitrarily large cubes where $E$ takes up at least a constant
fraction of the measure) then its points implement all sufficiently large
distances (conjecture by Székely [Szé83]).

Another well known problem, very much related to the contents of this paper,
is the so-called Erdős similarity problem: A set
${\mathbb{A}}\subseteq{\mathbb{R}}$ is called universal in measure if whenever
$E\subseteq{\mathbb{R}}$ has positive Lebesgue measure we can find an affine
copy of $A$ contained in $E$. In other words $x+t{\mathbb{A}}\subseteq E$ for
some $x\in{\mathbb{R}},t>0$. It is easy to see that every finite set
${\mathbb{A}}$ is universal (just look close enough to some point of density
of $E$, shrink ${\mathbb{A}}$ enough and average the number of points of the
copy of ${\mathbb{A}}$ that belong to $E$ over translates of ${\mathbb{A}}$
nearby) but it has been conjectured [Erd15] (see also [CFG12, p. 183]) that no
infinite set ${\mathbb{A}}$ can be universal in measure. This is known for
many classes of infinite sets but not for all [Fal84, GLW23, HL98, Kom83,
Chl15]. Clearly it would suffice to prove this for ${\mathbb{A}}$ being a
positive sequence $a_{n}$ decreasing to $0$ but if $a_{n}$ decays fast to 0
(so it is in some sense sparse, hence not that hard to contain) this is still
unknown. On the contrary this is known when $\log\frac{1}{a_{n}}=o(n)$. This
is not known if $a_{n}=2^{-n}$, for example.

In this paper we consider an analogue of the Erdős similarity problem “in the
large”. Let ${\mathbb{A}}\subseteq{\mathbb{R}}$ be a discrete, unbounded,
infinite set in ${\mathbb{R}}$. Can we find a “large” measurable set
$E\subseteq{\mathbb{R}}$ which does not contain any affine copy
$x+t{\mathbb{A}}$ of ${\mathbb{A}}$ (for any $x\in{\mathbb{R}},t>0$)? Our
attention to this problem was drawn by a recent paper by Bradford, Kohut and
Mooroogen [BKM23] in which the authors prove that if ${\mathbb{A}}$ is an
infinite arithmetic progression then this is indeed possible: for any
$p\in[0,1)$ they construct a Lebesgue measurable set $E$, with measure at
least $p$ in any interval of length 1, which does not contain any affine copy
of ${\mathbb{A}}$. This is clearly equivalent to being able to obtain, for any
$p\in[0,1)$ a set $E$ avoiding all infinite arithmetic progressions and having
measure $\geq p$ in any interval of length 1 whose endpoints are integers.
(Indeed, if the set $E$ has measure at least $p$ in every interval of the form
$[n,n+1]$, $n\in{\mathbb{Z}}$, then, since for any $x$ the interval $[x,x+1]$
is contained in the union of two such unit-length intervals with integer
endpoints, we obtain that $[x,x+1]\setminus E$ has measure at most $2(1-p)$.
Since $p$ can be as close to 1 as we want, this implies that $[x,x+1]\setminus
E$ has measure as close to 0 as we want.) From now on we follow this
simplification and we deal only with intervals with integer endpoints (in any
dimension).

We generalize the result of [BKM23] to sequences of nonnegative numbers
${\mathbb{A}}$ which do not grow too fast. To state our result, we introduce
the following class of sequences.

######  Definition 1.1.

We say that a real sequence ${\mathbb{A}}={\left\\{{a_{n},\
n\in{\mathbb{N}}}\right\\}}$ is in the class (A) if

  1. (1)

$a_{0}=0$,

  2. (2)

$a_{n+1}-a_{n}\geq 1$, for every $n\in\mathbb{N}$.

  3. (3)

$\log a_{n}=o(n)$

Remark. Since the problem we are studying is translation invariant condition
1.1.(1) in Definition 1.1 is unnecessary, but we keep it as it simplifies the
proofs somewhat.

Writing

(1.1) $A(t)={\left|{{\mathbb{A}}\cap[0,t]}\right|}$

for the counting function of the set ${\mathbb{A}}$, notice that the growth
condition 1.1.(3) is equivalent to the limit, as $t\to+\infty$,

(1.2) $\frac{A(t)}{\log t}\to+\infty.$

Our main result is the following.

######  Theorem 1.1.

Consider the sequence $\mathbb{A}=\\{a_{n}:\;n\in\mathbb{N}\\}$ which belongs
to the class (A). Then, for each $0\leq p<1$, there exists a Lebesgue
measurable set $E\subseteq\mathbb{R}$ such that

${\left|{E\cap[m,m+1]}\right|}\geq p,\ \ \ \text{ for all }m\in{\mathbb{Z}},$

but $E$ does not contain any affine copy of $\mathbb{A}$.

As in the case of the Erdős similarity problem described above, the sparser
the set ${\mathbb{A}}$ is the easier it should be to be contained in large
sets, so it is not surprising that we had to impose a growth condition (to
belong to the class (A)). It remains an open question if a similar set $E$ can
be constructed when ${\mathbb{A}}$ grows exponentially or faster.

######  Question 1.

Is there a sequence $0<a_{n}\to+\infty$ and a number $p\in[0,1)$ such that one
can find an affine copy of ${\mathbb{A}}={\left\\{{a_{n}:\
n\in{\mathbb{N}}}\right\\}}$ in any set $E\subseteq{\mathbb{R}}$ which has
measure more than $p$ in any interval of length 1?

Unlike the approach taken in [BKM23] our method of proof is probabilistic. We
construct a family of random sets and we show that, with high probability,
such a random set will have all the properties we want. This method turns out
to be extremely flexible, and this allows us to generalize. Not only can we
deal with essentially arbitrary and unstructured sequences ${\mathbb{A}}$ but
we can also relax the sense in which we seek copies of ${\mathbb{A}}$ in the
large set $E$. Instead of scaling the elements of ${\mathbb{A}}$ and
translating them

$x+ta_{n},\ \ x\in{\mathbb{R}},t>0,$

we can allow for more general transformations

(1.3) $x+\phi(n,t)\cdot a_{n},\ \ x\in{\mathbb{R}},t>0.$

######  Theorem 1.2.

Consider the set $\mathbb{A}=\\{a_{n}:\;n\in\mathbb{N}\\}$, which belongs to
the class (A), and let $\phi(n,t):{\mathbb{N}}\times(0,+\infty)\to(0,+\infty)$
be such that for each $n$ the function $\phi(n,t)$ is increasing in $t$ and is
such that for all $n\in{\mathbb{N}}$ we have

(1.4) $C_{1}t\leq\phi(n+1,t)a_{n+1}-\phi(n,t)a_{n}$

and

(1.5) $\phi(n,t)\leq C_{2}t,\ \ \text{ for all }t>0,$

for some $C_{1},C_{2}>0$. Then, for each $0\leq p<1$, there exists a Lebesgue
measurable set $E\subseteq\mathbb{R}$ such that $E$ intersects every interval
of unit length in a set of measure at least $p$, but $E$ does not contain the
set

${\left\\{{x+\phi(n,t)\cdot a_{n}:n\in{\mathbb{N}}}\right\\}}$

for any choice of $x\in{\mathbb{R}},t>0$.

We adopt certain arguments from [Kol97, Section 3] where it is proved, on the
Erdős similarity problem, that sequences with a finite limit, say $0$, which
are not decaying very fast (e.g. they decay polynomially or subexponentially
but not, for instance, exponentially fast – compare to our growth condition
(1.1.(3))), cannot be universal in measure, by showing the existence of a
randomly constructed set $E\subseteq[0,1]$, avoiding all affine copies of the
sequence.

The measure assumption makes this problem different than other “avoidance"
problems, where the avoiding set is often taken to have zero Lebesgue measure
but to have large Hausdorff dimension or Fourier dimension. For example, in
[Kel08], a compact subset of $\mathbb{R}$ is constructed that has full
Hausdorff dimension but does not contain any $3$-term arithmetic progression.
See also [CLP22, DPZ21, FP18, Mag11, Mát17, Shm17, Yav21].

We can also prove the following result in higher dimension. We phrase it as
avoiding linear images of a set in Euclidean space into another Euclidean
space. In this manner we obtain easily some corollaries, Theorem 1.1 one of
them, and its proof is rather simpler than that of Theorem 1.1 given in §3.
But it does not extend easily to more complicated transformations such as
those in Theorem 1.2, so we choose to stay with linear maps.

######  Theorem 1.3.

Let $d_{1},d\geq 1$, $b,f>0$, $p\in[0,1)$. Let also $\alpha(R)$ be a function
satisfying $\displaystyle\frac{\alpha(R)}{\log R}\to+\infty$ as $R\to+\infty$.

Then if ${\mathbb{A}}\subseteq{\mathbb{R}}^{d_{1}}$ is a discrete point set
such that

(1.6) ${\left|{{\mathbb{A}}\cap B_{R}(0)}\right|}\leq C_{2}R^{b},\ \ \ (R>0)$

there is a set $E\subseteq{\mathbb{R}}^{d}$ such that

  1. i.

${\left|{E\cap(m+[0,1]^{d})}\right|}\geq p$ for all $m\in{\mathbb{Z}}^{d}$,

  2. ii.

For any linear map $T:{\mathbb{R}}^{d_{1}}\to{\mathbb{R}}^{d}$ if for
arbitrarily large values of $R$

(1.7) $T({\mathbb{A}})\cap B_{R}(0)$

contains at least $\alpha(R)$ points with separation $R^{-f}$ then

(1.8) $T({\mathbb{A}})\text{ is not contained in }E.$

###### Proof of Theorem 1.1 using Theorem 1.3.

Apply Theorem 1.3 with $d_{1}=2,d=1$, $b=1,\alpha(x)=A(x^{1/2})$ (where $A(x)$
is the counting function of ${\mathbb{A}}$), $f=1$ (there is great flexibility
in choosing $\alpha(x),b,f$) and the set

$P={\mathbb{A}}\times{\left\\{{1}\right\\}}\subseteq{\mathbb{R}}^{2}$

to obtain a set $E\subseteq{\mathbb{R}}$ satisfying
${\left|{E\cap[m,m+1]}\right|}\geq p$ for all $m\in{\mathbb{Z}}$. We see that
(1.6) is satisfied. Let now $T:{\mathbb{R}}^{2}\to{\mathbb{R}}$ be given by
the $1\times 2$ matrix $T=(t,x)$ so that

$T(P)=x+t{\mathbb{A}}.$

For any $x\in{\mathbb{R}},t>0$, the set $(x+t{\mathbb{A}})\cap[-R,R]$ contains
at least $A(R/t)$ points of separation $t$, so, if $R$ is large enough, it
contains $\alpha(R)=A(R^{1/2})$ points with separation $R^{-1}$. It follows
that $x+t{\mathbb{A}}$ is not contained in $E$. ∎

######  Corollary 1.4 (Avoiding linear images of general sets in high
dimension).

Let $p\in[0,1)$, $d\geq 1$, $a_{n}\in{\mathbb{R}}^{d}$, for
$n\in{\mathbb{N}}$, with $\log{\left|{a_{n}}\right|}=o(n)$ and
${\left|{a_{n}-a_{n+1}}\right|}\geq 1$ for all $n\in{\mathbb{N}}$. Then there
is a set $E\subseteq{\mathbb{R}}^{d}$ such that for all $m\in{\mathbb{Z}}^{d}$
we have ${\left|{E\cap(m+[0,1]^{d})}\right|}\geq p$ and such that for all
$x\in{\mathbb{R}}^{d}$ and for all non-singular linear
$T:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ the set ${\left\\{{x+Ta_{n}:\
n\in{\mathbb{N}}}\right\\}}$ is not contained in $E$.

###### Proof.

Take ${\mathbb{A}}\subseteq{\mathbb{R}}^{2d}$ to be the set
$A\times\\{(\underbrace{1,0,\ldots,0}_{d})\\}$, where $A={\left\\{{a_{n}:\
n\in{\mathbb{N}}}\right\\}}$. Writing $A(s)=\\#(A\cap B_{s}(0))$ for the
counting function of $A$ we have $\displaystyle\frac{A(R)}{\log R}\to+\infty$.
Use Theorem 1.3 with $d_{1}=2d$, $b=1$, $\alpha(R)=A(R^{1/2})$, $f=1$. Let
$T:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ be non-singular,
$x\in{\mathbb{R}}^{d}$, and define the linear map
$S:{\mathbb{R}}^{2d}\to{\mathbb{R}}^{d}$ by

$S(u,v)=S(u,v_{1},v_{2},\ldots,v_{d})=Tu+v_{1}x.$

In other words the $d\times(2d)$ matrix of $S$ is $(T\ |\ x\ |\ 0)$ in block
form. It follows that

$S({\mathbb{A}})={\left\\{{Ta_{n}+x:\ n\in{\mathbb{N}}}\right\\}}.$

Since $T$ is non-singular it follows that if $R>0$ is sufficiently large the
set $S({\mathbb{A}})\cap B_{R}(0)$ contains at least $\alpha(R)$ points with
separation $\geq R^{-1}$ so the set $E\subseteq{\mathbb{R}}^{d}$ furnished by
Theorem 1.3 does not contain $S({\mathbb{A}})$, as we had to prove. ∎

######  Corollary 1.5 (Corollary 6 from [BKM23]).

If $p\in[0,1)$ then there exists a set $E\subseteq{\mathbb{R}}^{d}$ such that
${\left|{E\cap(m+[0,1]^{d})}\right|}\geq p$ for all $m\in{\mathbb{Z}}^{d}$ and
it does not contain any set of the form $x+{\mathbb{N}}\Delta$, with
$x\in{\mathbb{R}}^{d}$,
$\Delta\in{\mathbb{R}}^{d}\setminus{\left\\{{0}\right\\}}$ (an arithmetic
progression in ${\mathbb{R}}^{d}$).

###### Proof.

We use Corollary 1.4 with the sequence
$a_{n}=(n,0,\ldots,0)\in{\mathbb{R}}^{d}$, $x\in{\mathbb{R}}^{d}$ and any non-
singular $d\times d$ matrix $T$ that maps $(1,0,\ldots,0)$ to $\Delta$. ∎

The outline of this note is as follows. In §3 we give the proof of Theorem 1.1
without using Theorem 1.3, and we indicate how the same proof also works for
Theorem 1.2. In §4 we extend our technique to cover linear transformations of
given sequences from one Euclidean space to another and prove Theorem 1.3 and
some corollaries.

Added in revision: The results in [BGK+22], which came after this paper was
submitted, are very relevant to the results in this paper and contain some
improvements.

##  2\. Warm-up and some basic tools: no translational copies

In this section we introduce the basic probabilistic method by proving the
more restricted Theorem 2.1: we can avoid all translations of a given infinite
sequence $0\leq a_{n}\to+\infty$ with a set which is arbitrarily large
everywhere. This is considerably easier than avoiding all affine copies of the
sequence, when scaling the sequence as well as translating it is allowed. For
translations we have only one degree of freedom while for affine copies we
have two. Still, some important ingredients of the method will be evident in
the proof of Theorem 2.1 below. In §3 we will introduce the extra
discretization in scaling space that will be required.

######  Theorem 2.1.

Let
${\mathbb{A}}={\left\\{{a_{0}=0<a_{1}<a_{2}<\cdots}\right\\}}\subseteq{\mathbb{R}}$
be a sequence with $a_{n}\to+\infty$, and $p\in[0,1)$. Then we can find a
Lebesgue measurable set $E\subseteq{\mathbb{R}}$ such that no translate of
${\mathbb{A}}$

$x+{\mathbb{A}},\ \ x\in{\mathbb{R}},$

is contained in $E$, and such that for each $m\in{\mathbb{Z}}$ we have

${\left|{E\cap[m,m+1]}\right|}\geq p.$

###### Proof.

Let $q<1$ be defined by $1-q=\frac{1}{2}(1-p)$ (or $q=\frac{1}{2}(1+p)$).
Passing to a subsequence we can assume that $a_{n+1}-a_{n}\geq 1$ for all $n$.
We construct a random set $E$ by breaking up each unit interval $[m,m+1]$,
$m\in{\mathbb{Z}}$, into a number $N_{m}$ of equal intervals and keeping each
of these subintervals with probability $q$, independently, into our set $E$.
As ${\left|{m}\right|}$ increases the number $N_{m}$ will also have to
increase, so let us take $N_{m}=\max{\left\\{{K,{\left|{m}\right|}}\right\\}}$
say, where the large positive integer $K$ will be determined later.

unitsize(1.0cm);

int i, j, N=4; int M[]=4, 5, 6, 7, 8;

srand(123);

draw((-N-0.5, 0) – (N+0.5, 0)); for(i=-N; i<=N; ++i) //dot((i, 0));
draw((i,0.1)–(i,-0.05)); label(string(i), (i, 0), S); for(i=-N; i<N; ++i) int
m = i>=0?M[i]:M[-i-1]; real h=1/m, d=0.05;

for(j=0; j<m; ++j) draw((i+j*h, d) – (i+j*h, -d)); if(unitrand()>0.3)
draw((i+j*h, 0)–(i+(j+1)*h, 0), linewidth(4bp));

Figure 1. How the random set $E$ looks like

Define now for $x\in{\mathbb{R}}$ the random function

$\phi(x)={\mathds{1}}\left(x+{\mathbb{A}}\subseteq E\right).$

Since all points of $x+{\mathbb{A}}$ are in different random intervals it
follows, by independence, that
${\mathbb{E}}{\phi(x)}={{\mathbb{P}}\left[{x+{\mathbb{A}}\subseteq
E}\right]}=0$. Let the set of “bad” $x$ be

$B={\left\\{{x\in{\mathbb{R}}:\ x+{\mathbb{A}}\subseteq E}\right\\}}.$

We have

${\mathbb{E}}{{\left|{B}\right|}}=\int{\mathbb{E}}{\phi(x)}\,dx=0,$

hence ${\left|{B}\right|}$ is almost surely 0.

It remains to make sure that ${\left|{E\cap[m,m+1]}\right|}\geq p$ for all
$m\in{\mathbb{Z}}$. Fix $m$ and let $X_{1},\ldots,X_{N_{m}}$ be 0/1 random
variables such that $X_{i}$ is 0 if we included the $i$-th subinterval of
$[m,m+1]$ into the set $E$ and is 1 otherwise. In other words, $X_{i}$ denotes
the absence of the $i$-the subinterval from the set $E$. Clearly
${\mathbb{E}}{X_{i}}=1-q$ and the random variable

$X=\sum_{i=1}^{N_{m}}X_{i}\text{ (the number of missing subintervals) }$

is a sum of independent indicator random variables with
${\mathbb{E}}{X}=(1-q)N_{m}$ and we can use the very versatile large deviation
Chernoff inequality (to be used repeatedly in §§3.1,4 below)

(2.1)
${{\mathbb{P}}\left[{{\left|{X-{\mathbb{E}}{X}}\right|}\geq\epsilon{\mathbb{E}}{X}}\right]}\leq
2e^{-c_{\epsilon}{\mathbb{E}}{X}}$

(see [Che52, AS16]) with $\epsilon=1$ to obtain

$\displaystyle{{\mathbb{P}}\left[{{\left|{E\cap[m,m+1]}\right|}<p}\right]}$
$\displaystyle={{\mathbb{P}}\left[{X>(1-p)N_{m}}\right]}$
$\displaystyle={{\mathbb{P}}\left[{X-{\mathbb{E}}{X}>{\mathbb{E}}{X}}\right]}$
(2.2) $\displaystyle\leq
2\exp(-c_{1}(1-q)\max{\left\\{{K,{\left|{m}\right|}}\right\\}}).$

Define now the bad events
$B_{m}={\left\\{{{\left|{E\cap[m,m+1]}\right|}<p}\right\\}}$ which we want not
to hold, for all $m\in{\mathbb{Z}}$, and observe that the above inequality
means that we can choose $K$ large enough to achieve

$\sum_{m\in{\mathbb{Z}}}{{\mathbb{P}}\left[{B_{m}}\right]}<\frac{1}{2}.$

This means that with probability at least $1/2$ none of the bad events $B_{m}$
hold and, with the same probability, the set $B$ has measure 0. We now amend
our random set $E$ by removing from it the set $B$ (the set of first terms of
those $x+{\mathbb{A}}$ which are contained in $E$). Thus arises a set
$E^{\prime}$, which differs from $E$ by a set of measure $0$, and which
contains no translate of ${\mathbb{A}}$. ∎

######  Remark 2.1.

It is not necessary to assume that $a_{n}\to+\infty$ in Theorem 2.1. It
suffices to assume that the set ${\mathbb{A}}$ is infinite. If ${\mathbb{A}}$
does not contain a sequence tending to infinity (for Theorem 2.1 to apply to
it) then it will have a finite accumulation point, so a result of Komjáth
[Kom83] guarantees the existence of a set $\widetilde{E}\subseteq[0,1]$, of
measure arbitrarily close to $1$, which contains no translate of
${\mathbb{A}}$. Repeating $\widetilde{E}$ 1-periodically

$E=\bigcup_{n\in{\mathbb{Z}}}\widetilde{E}+n$

we obtain a set $E$ with the required properties. For a probabilistic proof of
this result in the spirit of the present paper see [Kol97].

######  Remark 2.2.

The Chernoff inequality (2.1) is extremely useful when one needs to control a
random variable $X$ (this means that one wants to ensure, with high
probability, that $X$ is near its mean ${\mathbb{E}}{X}$) which is a sum of
indicator, independent random variables. The key is that the mean
${\mathbb{E}}{X}$ cannot be very small, as it appears in the exponent in the
right hand side of (2.1). Since one usually wants to do so simultanesouly for
a large number of random variables $X$, one key situation to keep in mind is
the following: if the number of random variables to be controlled is
polynomial in $N$ (a parameter) it is enough that their mean are at least a
large multiple of $\log N$.

With minor modifications of the proof we can get a progressively denser set
$E$ avoiding all translates. We throw in the whole negative half line (as we
could have done in Theorem 1.1 too).

######  Theorem 2.2.

Let
${\mathbb{A}}={\left\\{{a_{0}=0<a_{1}<a_{2}<\cdots}\right\\}}\subseteq{\mathbb{R}}$
be a sequence with $a_{n}\to+\infty$. Then we can find a Lebesgue measurable
set $E\subseteq{\mathbb{R}}$ such that no translate of ${\mathbb{A}}$

$x+{\mathbb{A}},\ \ x\in{\mathbb{R}},$

is contained in $E$, and such that

$(-\infty,0]\subseteq E\ \text{ and }\ {\left|{E\cap[m,m+1]}\right|}\to
1^{-}\text{ as }m\to+\infty.$

###### Proof.

We indicate the differences with the proof of Theorem 2.1 and omit some
details.

Our random set $E$ now will be of the same type as in the proof of Theorem 2.1
but with the probability of including the small subintervals tending slowly to
$1$ as we go out to $+\infty$ and with the negative half line contained in $E$
to begin with.

Let us view the probability of keeping an interval as a function $p(s)$
defined on the real line. In the proof of Theorem 2.1 this function was
constant. Here it will be constant on all intervals of the form $[m,m+1]$,
$m\in{\mathbb{Z}}$.

With $\phi(x)={\mathds{1}}\left(x+{\mathbb{A}}\subseteq E\right)$ we need
again to ensure that ${\mathbb{E}}{\phi(x)}=0$ for all $x\in{\mathbb{R}}$.
After assuming, as in the previous proof, that the points of ${\mathbb{A}}$
differ by at least 1, we again have independence of all events $x+a\in E$ for
$a\in{\mathbb{A}}$ so that ${\mathbb{E}}{\phi(x)}=0$ becomes equivalent to

$\prod_{a\in{\mathbb{A}}}p(x+a)=0,$

which, writing $q(s)=1-p(s)$, is equivalent to

(2.3) $\sum_{a\in{\mathbb{A}}}q(x+a)=+\infty.$

Let $0=k_{1}<k_{2}<\cdots$ be those positive integers for which

$[k,k+1)\cap{\mathbb{A}}\neq\emptyset.$

Define then $q(x)$ to be $1/i$ in the interval $[{k_{i}},{k_{i+1}})$,
$i=1,2,\ldots$. It follows easily that for all $x\in{\mathbb{R}}$ we have
(2.3): since the function $q(\cdot)$ is decreasing we have $q(x+a_{n})\geq
q(a_{n})$ if $x\leq 0$ and if $x\geq 0$ we have $q(x+a_{n})\geq
q(a_{{\left\lceil{x}\right\rceil}+n})$ since $a_{k+1}-a_{k}\geq 1$ for all
$k\in{\mathbb{N}}$. In both cases the series (2.3) contains a tail of the
series $\sum_{a\in{\mathbb{A}}}q(a)$ which is divergent.

It remains to ensure that the random variables ${\left|{[m,m+1]\setminus
E}\right|}$ tend to 0 with ${m}\to+\infty$. These random variables are
$\frac{1}{N_{m}}$ times a sum of independent indicator random variables (one
for each of the $N_{m}$ subintervals into which we break up $[m,m+1]$) of mean
$q(m)N_{m}$ so we can use the Chernoff bound (2.1) to obtain

${{\mathbb{P}}\left[{{\left|{[m,m+1]\setminus E}\right|}>2q(m)}\right]}\leq
2\exp(-c_{1}q(m)N_{m}).$

To ensure that the sum, over all $m\in{\mathbb{Z}}$ of the left hand side is
$<1$ we can of course pick the integers $N_{m}$ to be very large, say
$N_{m}=K\frac{1}{q(m)}{\left|{m}\right|}$, with a sufficiently large constant
$K>0$.

∎

##  3\. No affine copies for slowly increasing sequences

In this section we prove Theorem 1.1 and explain why the proof also gives the
more general Theorem 1.2.

######  Lemma 3.1.

Let $\mathbb{A}\in(A)$. For all $0<a<b$, $0\leq p<1$ and $\epsilon>0$, there
is $N_{0}\in\mathbb{N}$, such that for all $N\geq N_{0}$, there is a set
${E\subseteq[-N,N]}$ such that

  1. (i)

for all $m\in\\{-N,-N+1,...,N-1\\}$, we have
${\left|{E\cap[m,m+1]}\right|}\geq p$, and

  2. (ii)

if the set $B$ consists of all $x\in[-N,N]$ for which there is $t\in[a,b]$
such that

(a) $(x+t\mathbb{A})\cap[-N,N]\subseteq E$ and

(b) $\\#((x+t\mathbb{A})\cap[-N,N])\geq A\left(\frac{N}{10b}\right)$,

then ${\left|{B}\right|}<\epsilon$. Here, $A(\cdot)$ is the counting function
(1.1) of the set ${\mathbb{A}}$ and
$A\left(\frac{N}{10b}\right)={\left|{{\mathbb{A}}\cap[0,N/(10b)]}\right|}$.

Let us first show how one derives Theorem 1.1 from Lemma 3.1. We give the
proof of Theorem 1.1 in two steps: the first one verifies the result for a
restricted scale, that is, for scales in a compact interval, and the second
one concludes for all positive scales, by writing the whole scaling interval
$(0,+\infty)$ as a countable union of intervals of the above type.

Step 1. For all $0<a<b$ and for each $0\leq p<1$, there exists a set
$E\subseteq\mathbb{R}$, such that ${\left|{E\cap[m,m+1]}\right|}\geq p$ for
all $m\in\mathbb{Z}$, but $E$ does not contain any affine copies of
$\mathbb{A}$ with scale in $[a,b]$.

Consider $0\leq p<1$ and a positive increasing sequence $\\{p_{n}\\}$,
$n=1,2,...$ such that $p_{n}\rightarrow 1^{-}$ and, moreover,

(3.1) $\sum\limits_{n=0}^{\infty}(1-p_{n})<1-p.$

Take also any positive sequence $\epsilon_{n}\rightarrow 0$. According to
Lemma 3.1, for $0<a<b$, we can choose an increasing sequence of natural
numbers $N_{n}=N_{n}(p_{n},\epsilon_{n},a,b)\to\infty$, for which there exist
sets $E_{n}\subseteq[-N_{n},N_{n}]$ with the following properties:

  * (i)

for all $m=-N_{n},...,N_{n}-1$, we have ${\left|{E_{n}\cap[m,m+1]}\right|}\geq
p_{n}$,

  * (ii)

if

${\mathbb{A}}_{n}(x,t)=(x+t\mathbb{A})\cap[-N_{n},N_{n}]$

and

$B_{n}=\\{x\in[-N_{n},N_{n}]:\;\exists t\in[a,b]\text{ s.t.
}{\mathbb{A}}_{n}(x,t)\subseteq E_{n}$ $\text{ and
}\\#{\mathbb{A}}_{n}(x,t)\geq A\left(\frac{N_{n}}{10b}\right)\\},$

then ${\left|{B_{n}}\right|}<\epsilon_{n}$.

Now take

$\widetilde{E_{n}}=\left(-\infty,-N_{n}\right]\cup
E_{n}\cup\left[N_{n},+\infty\right)$

and

$E=\bigcap\limits_{n=1}^{\infty}\widetilde{E_{n}}.$

size(9cm,0); //unitsize(1.6cm);

int i, j, N=4; //int M[]=4, 5, 6, 7, 8; int m = 6; real h=1/m, d=0.05;

srand(1234);

draw((-N-0.5, 0) – (N+0.5, 0)); for(i=-N; i<=N; ++i) //dot((i, 0));
draw((i,0.1)–(i,-0.05)); label(abs(i)<N?".":(i>0?"$\cdots
N_{n}$":"$-N_{n}\cdots$"), (i, 0), S); if(abs(i)==N) draw((i, 0.2)–(i,-0.05));
label("$0$", (0, -d), S);

for(i=-N; i<N; ++i) for(j=0; j<m; ++j) draw((i+j*h, d) – (i+j*h, -d));
if(unitrand()>0.3) draw((i+j*h, 0)–(i+(j+1)*h, 0), linewidth(4bp));

draw((N, 0)–(2*N, 0), linewidth(4bp)); draw((-N, 0)–(-2*N, 0),
linewidth(4bp));

Figure 2. The set $\widetilde{E_{n}}$.

Then, since ${\left|{\widetilde{E_{n}}\cap[m,m+1]}\right|}\geq p_{n}$ for all
$m\in\mathbb{Z}$, we get from (3.1) that the set $E$ has measure at least $p$
at every unit interval with integer endpoints. Also, if there exist $x$, $t$
such that $x+t\mathbb{A}\subseteq E$, then $x+t\mathbb{A}$ is also contained
in each $\widetilde{E_{n}}$. Having fixed $x$ and $t$ we can then find $n_{0}$
large enough such that for all $n\geq n_{0}$, we have
$\\#((x+t\mathbb{A})\cap[-N_{n},N_{n}])\geq A\left(\frac{N_{n}}{10b}\right)$.
This implies that for every $n\geq n_{0}$, $x\in B_{n}$. It follows that for
every $n\geq n_{0}$, ${\left|{B_{n}}\right|}<\epsilon_{n}$. Since
$\epsilon_{n}\rightarrow 0$, setting

$B=\\{x:\;\exists t\in[a,b]\text{ s.t. }x+t\mathbb{A}\subseteq E\\},$

we get ${\left|{B}\right|}=0$. The null set of “bad" translates $B$ is
contained in $E$ (since we assumed that $0\in{\mathbb{A}}$), thus removing it
from $E$ results in a set $E^{\prime}$, which still has measure
${\left|{E^{\prime}\cap[m,m+1]}\right|}\geq p$ for all $m\in\mathbb{Z}$, but
contains no affine copy of $\mathbb{A}$ with scale in $[a,b]$.

Step 2. Completion of the proof of Theorem 1.1.

Take a positive sequence $p_{n}^{\prime}\in[0,1)$, $n\in\mathbb{Z}$, such that

(3.2) $\sum\limits_{n\in\mathbb{Z}}(1-p_{n}^{\prime})<{1-p}.$

Consider the intervals $[a_{n},b_{n}]=[2^{n-1},2^{n}]$, $n\in\mathbb{Z}$.
Then, according to Step 1, for each $p_{n}^{\prime}$, there exists a set
$E_{n}$ such that ${\left|{E_{n}\cap[m,m+1]}\right|}\geq p_{n}^{\prime}$, for
all $m\in\mathbb{Z}$, but for all $x\in\mathbb{R}$ and for all
$t\in[a_{n},b_{n}]$, the set $x+t\mathbb{A}$ is not contained in $E_{n}$.

Take

$E=\bigcap_{n\in\mathbb{Z}}E_{n}.$

Assume that for some $x\in\mathbb{R}$ and some $t>0$, $x+t\mathbb{A}\subseteq
E$. Then, $x+t\mathbb{A}\subseteq E_{n}$, for all $n\in\mathbb{Z}$. However,
since there is $n_{0}\in{\mathbb{Z}}$ such that $t\in[2^{n_{0}-1},2^{n_{0}}]$,
the inclusion $x+t\mathbb{A}\subseteq E_{n_{0}}$ cannot be true. Thus, $E$
does not contain any affine copy of $\mathbb{A}$ with positive scale. Finally,
due to (3.2) we have ${\left|{[m,m+1]\setminus E}\right|}<1-p$, or
${\left|{E\cap[m,m+1]}\right|}\geq p$.

###  3.1. Proof of Lemma 3.1

Fix the scale $t\in[a,b]$ and let $0\leq p<1$. Consider the positive sequence
given by

(3.3)
$p_{N}=1-{\sqrt{\frac{\log\left(\frac{N}{10b}\right)}{A\left(\frac{N}{10b}\right)}}}.$

From (1.2) this implies $p_{N}\rightarrow 1^{-}$.

Partition $[-N,N]$ into unit intervals $[m,m+1]$, $m=-N,-N+1,...,N-1$. Divide
each $[m,m+1]$ further, into $k_{N}$ equal subintervals

$I_{i,m}=m+\left[\frac{i-1}{k_{N}},\frac{i}{k_{N}}\right],\quad
i=1,...,k_{N},$

where

(3.4) $k_{N}=\left\lceil\frac{10}{a}\right\rceil\frac{N}{1-p_{N}}.$

Notice that $k_{N}/N\to+\infty$.

Construct a random set $E=E_{N}$ as follows: keep each $I_{i,m}$ in $E$
independently of the other intervals and with probability $p_{N}$ as in (3.3).
Then, $\mathbb{P}(x\in E)=p_{N}$ for each $x\in[-N,N]$.

unitsize(1.0cm);

int i, j, N=4; //int M[]=4, 5, 6, 7, 8;

srand(1234);

draw((-N-0.5, 0) – (N+0.5, 0)); for(i=-N; i<=N; ++i) //dot((i, 0));
draw((i,0.1)–(i,-0.05)); label(abs(i)<N?string(i):(i>0?"$\cdots
N$":"$-N\cdots$"), (i, 0), S); for(i=-N; i<N; ++i) int m = 6; real h=1/m,
d=0.05;

for(j=0; j<m; ++j) draw((i+j*h, d) – (i+j*h, -d)); if(unitrand()>0.3)
draw((i+j*h, 0)–(i+(j+1)*h, 0), linewidth(4bp));

Figure 3. The random set $E$.

Let $M_{N}(x,t)$ be the number of elements of $(x+t\mathbb{A})\cap[-N,N]$ and
observe that

(3.5) $M_{N}(x,t)\leq A(2N/a),\ \ \text{ for }x\in[-N,N].$

For a given set $E\subseteq[-N,N]$, consider the set of “bad" translates

$B=\Bigl{\\{}x\in[-N,N]:\;\exists t\in[a,b]\text{ s.t. }\\\
(x+t\mathbb{A})\cap[-N,N]\subseteq E$ $\text{ and }M_{N}(x,t)\geq
A\left(\frac{N}{10b}\right)\Bigr{\\}}.$

We first deal with the measure of $B$. We have

$\displaystyle\mathbb{E}{\left|{B}\right|}$
$\displaystyle=\mathbb{E}\int_{-N}^{N}\mathbbm{1}_{B}(x)dx$ (3.6)
$\displaystyle=\int_{-N}^{N}\mathbb{P}\Bigl{[}\exists
t\in[a,b]:\;(x+t\mathbb{A})\cap[-N,N]\subseteq E$ (3.7) $\displaystyle\text{ \
\ \ \ \ \ \ \ \ and }M_{N}(x,t)\geq A\left(\frac{N}{10b}\right)\Bigr{]}\,dx.$

In what follows, we estimate from above the probability in (3.7), uniformly in
$x\in[-N,N]$.

Fix $x\in[-N,N]$. To check whether there exists $t\in[a,b]$ such that
$(x+t\mathbb{A})\cap[-N,N]\subseteq E$, it is sufficient to check whether such
a $t$ exists in a finite set

(3.8) $S=S(x)=\\{t_{1},t_{2},...,t_{u}\\}\subseteq[a,b].$

unitsize(1.0cm);

int i, j, NN=4; int m = 6; real x, h=1/m, d=0.05;

srand(1234);

draw((-NN-0.5, 0) – (NN+0.5, 0)); for(i=-NN; i<=NN; ++i) //dot((i, 0));
draw((i,0.1)–(i,-0.05)); label(abs(i)<NN?string(i):(i>0?"$\cdots
N$":"$-N\cdots$"), (i, 0), S); for(i=-NN; i<NN; ++i) for(j=0; j<m; ++j)
draw((i+j*h, d) – (i+j*h, -d)); if(unitrand()>0.3) draw((i+j*h, 0)–(i+(j+1)*h,
0), linewidth(4bp));

x = -2-h; draw((x, -d)–(x, d)); label((x, d), "$x$", NW);

draw((x,0)..(1,0.5)..(2,0), dashed, ArcArrow(HookHead)); label((1, 0.5),
"$x+t_{1}a_{n_{1}}$", N); draw((2,4*d)–(2.5,4*d), ArcArrow);

draw((x,0)..(1,1.5)..(3+2*h,0), dashed, ArcArrow(HookHead)); label((1, 1.5),
"$x+t_{2}a_{n_{2}}$", N); draw((3+2*h,4*d)–(3.5+2*h,4*d), ArcArrow);

Figure 4. As $x$ is held fixed and $t$ grows the points $x+ta_{n}$ cross over
interval endpoints creating events that need to be checked.

Write
$\alpha_{0}^{\prime}<\alpha_{1}^{\prime}<...<\alpha_{M_{N}(x,t)-1}^{\prime}$
for the elements of $(x+t\mathbb{A})\cap[-N,N]$. Then, the set $S$ consists
exactly of those $t\in[a,b]$ for which some $\alpha_{j}^{\prime}=x+ta_{j}$,
$j=0,...,M_{N}(x,t)-1$, is in the set
$m+\left\\{0,\frac{1}{k_{N}},\frac{2}{k_{N}},...,\frac{k_{N}-1}{k_{N}},1\right\\}$,
for some $m\in\\{-N,-N+1,...,N-1\\}$. Each of the points
$\alpha_{j}^{\prime}=x+ta_{j}$ traverses, as $t$ moves from $a$ to $b$, and as
long as the point $\alpha_{j}^{\prime}$ remains in $[-N,N]$, an interval of
length at most $2N$, therefore it meets at most $2Nk_{N}$ interval endpoints
of the intervals $I_{i,m}$. Altogether, we have

(3.9) $u\leq 2Nk_{N}\sup_{a\leq t\leq b}M_{N}(x,t)\leq
c(a)N^{2}(1-p_{N})^{-1}A\left(\frac{2N}{a}\right),$

where for the last inequality, we used (3.4) and (3.5).

Since $k_{N}\to+\infty$, we can take $N$ large enough, say $N\geq N_{0}$, so
that $k_{N}>1/a$, for every $N\geq N_{0}$. Then, the length of each $I_{i,m}$
is small enough, $\leq a$, to ensure that, for each $t\in[a,b]$, the points
$\alpha_{j}^{\prime}$, $j=0,...,M_{N}-1$, all belong to different intervals
$I_{i,m}$. Therefore, for any fixed $x$ and $t$,

$\displaystyle{{\mathbb{P}}\left[{(x+t\mathbb{A})\cap[-N,N]\subseteq E\text{
and }M_{N}(x,t)\geq A\left(\frac{N}{10b}\right)}\right]}$ $\displaystyle\ \
\leq{{\mathbb{P}}\left[{(x+t\mathbb{A})\cap[-N,N]\subseteq E\ \mid\
M_{N}(x,t)\geq A\left(\frac{N}{10b}\right)}\right]}$ (3.10) $\displaystyle\ \
\leq p_{N}^{A\left(\frac{N}{10b}\right)}.$

Thus, using the bound (3.9),

$\displaystyle{{\mathbb{P}}\left[{\exists t\in
S:\;(x+t\mathbb{A})\cap[-N,N])\subseteq E}\right]}$ $\displaystyle\ \ \ \leq
c(a)N^{2}(1-p_{N})^{-1}A\left(\frac{2N}{a}\right)p_{N}^{A\left(\frac{N}{10b}\right)}.$

Thus, (3.7) yields

$\mathbb{E}{\left|{B}\right|}\leq
2c(a)N^{3}(1-p_{N})^{-1}A\left(\frac{2N}{a}\right)p_{N}^{A\left(\frac{N}{10b}\right)}.$

We want to have

$N^{3}(1-p_{N})^{-1}A\left(\frac{2N}{a}\right)p_{N}^{A\left(\frac{N}{10b}\right)}\rightarrow
0,$

while $p_{N}\rightarrow 1^{-}$, as $N\rightarrow\infty$. Since $A(\cdot)$
grows at most linearly at infinity, it suffices to show that

(3.11) $\displaystyle A\left(\frac{N}{10b}\right)$ $\displaystyle\log
p_{N}\left(4\frac{\log N}{A\left(\frac{N}{10b}\right)\log
p_{N}}-\frac{\log(1-p_{N})}{A\left(\frac{N}{10b}\right)\log p_{N}}+1\right)$
$\displaystyle\rightarrow-\infty.$

To show (3.11), observe first that since
$\lim_{x\rightarrow+\infty}x\log\left(1-x^{-1/2}\right)=-\infty,$ we have

(3.12) $\frac{A\left(\frac{N}{10b}\right)\log p_{N}}{\log
N}\rightarrow-\infty,$

due to (3.3). Therefore, we also have $A\left(\frac{N}{10b}\right)\log
p_{N}\rightarrow-\infty$. Finally, by (3.3) and (3.12) we get

$\frac{\log(1-p_{N})}{A\left(\frac{N}{10b}\right)\log
p_{N}}=-\frac{1}{2}\frac{\log
A\left(\frac{N}{10b}\right)}{A\left(\frac{N}{10b}\right)\log
p_{N}}\left\\{1-\frac{\log\log\frac{N}{10b}}{\log
A\left(\frac{N}{10b}\right)}\right\\}\rightarrow 0.$

In other words, we have shown that for every $\epsilon>0$, there is $N_{1}\geq
N_{0}$ such that for all $N\geq N_{1}$,
$\mathbb{E}{\left|{B}\right|}<\epsilon/2$, which implies that

(3.13) $\mathbb{P}({\left|{B}\right|}\geq\epsilon)<1/2,\quad\forall N\geq
N_{1}.$

We now turn to the measure of $E$ in every unit interval with integer
endpoints. Fix $m\in[-N,N]$. Let $X_{1}^{m},X_{2}^{m},...,X_{k_{N}}^{m}$ be
independent indicator random variables, with $X_{i}^{m}=1$ if and only if
$I_{i,m}\subseteq E$. Let $Y_{i}^{m}=1-X_{i}^{m}$ and denote by
$X^{m}=\sum_{i=1}^{k_{N}}X_{i}^{m}$, $Y^{m}=\sum_{i=1}^{k_{N}}Y_{i}^{m}$ their
sums. Then, $\mathbb{E}Y^{m}=(1-p_{N})k_{N}$. Notice also that the total
measure kept in $[m,m+1]\cap E$ is equal to $X^{m}/k_{N}$.

For any $\delta>0$ we define the “bad" events

$A_{m}=\\{|Y^{m}-\mathbb{E}Y^{m}|>\delta\,\mathbb{E}Y^{m}\\},\quad
m=-N,-N+1,...,N-1.$

To control ${{\mathbb{P}}\left[{A_{m}}\right]}$, we use Chernoff’s inequality,
[AS16, Che52]: for all $\delta>0$,

${{\mathbb{P}}\left[{A_{m}}\right]}\leq 2e^{-c_{\delta}\mathbb{E}Y^{m}},$

where
$c_{\delta}=\min{\left\\{{(1+\delta)\log(1+\delta)-\delta\log\delta,\,\delta^{2}/2}\right\\}}$.
Take $\delta=1/2$. It follows that

$\displaystyle{{\mathbb{P}}\left[{|Y^{m}-(1-p_{N})k_{N}|>\frac{1}{2}(1-p_{N})k_{N}}\right]}$
$\displaystyle\ \ \ \ \ \leq 2\exp\left(-{\frac{1}{2}(1-p_{N})k_{N}}\right).$

Thus, the probability that there is some $[m,m+1]\subseteq[-N,N]$, such that
$A_{m}$ holds, is at most

$4N\exp\left(-{\frac{1}{2}(1-p_{N})k_{N}}\right)$

and the right hand side tends to zero as $N\rightarrow+\infty$, by our choice
of $k_{N}$ in (3.4). Thus, there is $N_{2}\geq N_{1}$ such that

(3.14) ${{\mathbb{P}}\left[{\exists m\in\\{-N,-N+1,...,N-1\\}:\,A_{m}\text{
holds}}\right]}<\frac{1}{2},$

for all $N\geq N_{2}$. Then, (3.13) and (3.14) imply the existence of a set
$E\subseteq\mathbb{R}$ such that, on the one hand, it satisfies

${\left|{B}\right|}<\epsilon$

and on the other hand,

$X^{m}-p_{N}k_{N}\geq-\frac{1}{2}(1-p_{N})k_{N},$

for all $m=-N,-N+1,...,N-1$, for all $N\geq N_{2}$. Thus the measure of $E$ in
each unit interval $[m,m+1]$, is at least
$p_{N}-\frac{1}{2}(1-p_{N})\rightarrow 1$, as $p_{N}\rightarrow 1^{-}$. In
other words, for all $0\leq p<1$, there is $N_{3}\geq N_{2}$ such that for all
$N\geq N_{3}$, we have ${\left|{E\cap[m,m+1]}\right|}\geq p$. The proof of
Lemma 3.1 is now complete.

######  Remark 3.1.

Let us indicate here why the proof of Theorem 1.1 just completed also applies
to Theorem 1.2 without any essential changes. First of all, the implication
from Lemma 3.1 to Theorem 1.3 (finite to infinite) remains true almost
verbatim. So it suffices to ensure that Lemma 3.1 is true in this case. The
main ingredients of the proof of Lemma 3.1 are the following. Having fixed $x$
and varying $t$ we have to make sure that the following conditions hold.

  1. C.1

All points of the the $(x,t)$-copy of the set remain well separated, so that
independence applies and we can multiply the probabilities that they belong to
our random set. This is ensured by (1.4).

  2. C.2

The number of points in the $(x,t)$-copy of the set in the interval $[-N,N]$
has to be large as this is the exponent in the upper bound (3.10). Condition
(1.5) guarantees this.

  3. C.3

The number of events that need to be checked so that we are certain that for
all $t$ no $(x,t)$-copy is contained in our random set is small. This is the
number $u$ in (3.8). What we are doing in the proof is to count how many times
each of the points of our set (as $x$ is held fixed and $t$ increases from $a$
to $b$) crosses over an interval boundary. Since the $\phi(n,t)$ are assumed
increasing in $t$ this remains as before.

It should be clear that the conditions imposed on the scaling functions
$\phi(n,t)$ in Theorem 1.2 are far from optimal. They are rather indicative of
what can be accomplished with the method and it is clear that the method could
work under different sorts of conditions.

##  4\. The problem in higher dimension

We will derive Theorem 1.3 as a consequence of the more finitary theorem
below.

######  Theorem 4.1.

Let $d_{1},d\geq 1$, $\beta,\zeta>0$, $p\in(0,1)$. Let also $\alpha(N)$ be a
function satisfying $\displaystyle\frac{\alpha(N)}{\log N}\to+\infty$.

Then if $N$ is sufficiently large and $P\subseteq{\mathbb{R}}^{d_{1}}$ is a
point set with at most $N^{\zeta}$ points there is a set
$E_{N}\subseteq[-N,N]^{d}$ such that

  1. (1)

${\left|{E_{N}\cap\left(m+[0,1]^{d}\right)}\right|}\geq p$ for all
$m=(m_{1},\ldots,m_{d})\in{\mathbb{Z}}^{d}$, with $-N\leq m_{j}<N$,

  2. (2)

For any linear map $T:{\mathbb{R}}^{d_{1}}\to{\mathbb{R}}^{d}$ if

(4.1) $T(P)\cap[-N,N]^{d}$

contains at least $\alpha(N)$ points with separation$\geq N^{-\beta}$ then

(4.2) $\left(T(P)\cap[-N,N]^{d}\right)\subsetneq E_{N}.$

###### Proof.

Let $\gamma>\beta$ and split the cube $[-N,N]^{d}$ with a
$N^{-\gamma}\times\cdots\times N^{-\gamma}$-spaced grid of $O(dN^{1+\gamma})$
hyperplanes perpendicular to the $d$ coordinate axes. Define the random set
$E$ to contain each of the $N^{-\gamma}\times\cdots\times N^{-\gamma}$-sized
cubes independently with probability $p^{\prime}\in(p,1)$. We show that with
positive probability one can take $E_{N}=E$.

The first property of $E$ is a simple consequence of Chernoff bounds and we
can assume it holds with probability $>\frac{1}{2}$ working as in the proof of
Theorem 1.1.

Let $T=(T_{i,j})$ be a linear map ${\mathbb{R}}^{d_{1}}\to{\mathbb{R}}^{d}$.
This depends on $d\cdot d_{1}$ real variables $T_{i,j}$, so we view $T$ as an
element of ${\mathbb{R}}^{d\cdot d_{1}}$. Instead of checking condition (2)
for all $T\in{\mathbb{R}}^{d\cdot d_{1}}$ we first show that there is a small
number (polynomial in $N$) of $T$’s that need to be checked.

Indeed, the set of $N^{-\gamma}\times\cdots\times N^{-\gamma}$-sized cubes
that contain $T(P)$ does not change when $T$ varies except when one or more of
the points in $T(P)$ cross a dividing hyperplane of those that subdivide
$[-N,N]^{d}$. Let $H$ be one of those $O(dN^{1+\gamma})$ hyperplanes and fix
an arbitrary point $h\in H$. Let also $u$ be a unit vector orthogonal to $H$.
For a point $x\in{\mathbb{R}}^{d}$ to belong to $H$ it must satisfy the linear
equation

$E(H,x):\ \ u\cdot x=u\cdot h.$

Let $q\in P$. For the point $T(q)$ to belong to $H$ we must have

$E(H,T(q)):\ \ u\cdot T(q)=u\cdot h,$

which is a linear equation in $T\in{\mathbb{R}}^{d\cdot d_{1}}$. Taking all
such equations in $T$, over all dividing hyperplanes $H$ and all $q\in P$ we
obtain a subdivision of ${\mathbb{R}}^{d\cdot d_{1}}$ by

$n=O(d\cdot N^{1+\gamma}\cdot{\left|{P}\right|})$

hyperplanes. These $n$ hyperplanes subdivide ${\mathbb{R}}^{d\cdot d_{1}}$
into $m=O(n^{d\cdot d_{1}})$ connected regions (this is easily proved by
induction on the dimension, or see [Buc43]). For any two points $T_{1},T_{2}$
in the same region condition (4.2) is either true for both or false for both
since we can move continuously from $T_{1}$ to $T_{2}$ without leaving the
region and, therefore, without any of the point $T(q)$ touching any of the
dividing hyperplanes $H$.

size(9cm,0); //unitsize(1.6cm);

draw( (0, 0) – (10, 1) ); draw( (0, -2) – (10, 3) ); draw( (0, 8) – (10, -3)
); draw( (2, -2) – (3, 9) ); draw( (0, 5) – (10, 4) ); pair p1=(5,3), p2=(7.5,
2.3); dot("$T_{1}$", p1, N); dot("$T_{2}$", p2, N); draw(p1–p2, dashed); clip(
box((0,0), (10, 8)) );

Figure 5. The regions defined in $T$-space by the equations $E(H,T(q))$ for
all $H,q$. Only one of the transformations $T_{1}$, $T_{2}$ needs to be
checked.

It suffices therefore to check condition (4.2) for one point per region. Let
us call these points $T_{1},\ldots,T_{m}$. To guarantee that (4.2) holds for
all $T$ it is enough for it to be true for all $T_{j}$, $j=1,2,\ldots,m$.
Define the bad events

$B_{j}=\bigcap_{q\in P}{\left\\{{T_{j}(q)\in E}\right\\}}.$

We need to ensure that none of the $B_{j}$ holds, but we only need to check
those $B_{j}$ for which there is a $T$ in the cell of $T_{j}$ for which (4.1)
holds. For such a $j$ the number of different $N^{-\gamma}\times\cdots\times
N^{-\gamma}$-sized cubes touched by $T_{j}(P)$ is the same as the number
touched by $T(P)$ which is at least $\alpha(N)$ so

${{\mathbb{P}}\left[{B_{j}}\right]}\leq p^{\prime\alpha(N)},$

and it is therefore enough to make sure that

$n^{d\cdot d_{1}}p^{\prime\alpha(N)}=O\left(N^{\zeta\cdot d\cdot
d_{1}}N^{(1+\gamma)d\cdot d_{1}}p^{\prime\alpha(N)}\right)$

can be made arbitrarily small by choosing $N$ large. This is clearly possible
since the term $p^{\prime\alpha(N)}$ decays faster than any power of $N$. ∎

###### Proof of Theorem 1.3.

Let $p_{n}\in(0,1)$ be such that

(4.3) $\sum_{n=1}^{\infty}(1-p_{n})<1-p.$

Apply Theorem 4.1 successively for $N=n$, $p_{n}$, $\zeta=b$,
$\alpha(N)=\alpha(R)$, $\beta=f$ and the set
$P={\mathbb{A}}\cap[-n,n]^{d_{1}}$ to obtain sets $E_{n}\subseteq[-n,n]^{d}$.
Define

$E=\bigcap_{n=1}^{\infty}\left(E_{n}\cup({\mathbb{R}}^{d}\setminus[-n,n]^{d})\right).$

It is easy to see because of (4.3) that for any $m\in{\mathbb{Z}}^{d}$ we have
${\left|{E\cap m+[0,1]^{d}}\right|}\geq p$. Let
$T:{\mathbb{R}}^{d_{1}}\to{\mathbb{R}}^{d}$ and let $R$ be such that
$T({\mathbb{A}})\cap B_{R}(0)$ contains $\alpha(R)$ points which are $R^{-f}$
separated. Let $n={\left\lceil{R}\right\rceil}$. It follows from Theorem 4.1
that $T({\mathbb{A}})\cap[-n,n]^{d}$ is not contained in
$E_{n}\cup({\mathbb{R}}^{d}\setminus[-n,n]^{d})$ and therefore not contained
in $E$, as we had to show. ∎

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# Symmetry-induced decoherence-free subspaces

Jonathan Dubois  Ulf Saalmann  Jan Michael Rost  Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany

###### Abstract

Preservation of coherence is a fundamental yet subtle phenomenon in open
systems. We uncover its relation to symmetries respected by the system
Hamiltonian and its coupling to the environment. We discriminate between local
and global classes of decoherence-free subspaces for many-body systems through
the introduction of “ghost variables”. The latter are orthogonal to the
symmetry and the coupling to the environment does not depend on them.
Constructing them is facilitated in classical phase space and can be
transferred to quantum mechanics through the equivalent role that Poisson and
Lie algebras play for symmetries in classical and quantum mechanics,
respectively. Examples are given for an interacting spin system.

#### Introduction:

A physical system interacting with an environment Gardiner1983 ; Risken1984 ;
Gotze1992 ; Breuer2002  relaxes to an equilibrium or a non-equilibrium steady
state on time scales much longer than the relaxation time Qian2006 . In the
steady state the system no longer evolves in time due to its coupling to the
environment and _all_ associated physical observables reach a constant value.
This characteristic is inherent to classical and quantum open systems.

Quantum mechanically, relaxation is accompanied by decoherence which remains a
major obstacle in putting physical devices to work for quantum computation
Grover1997 ; Ladd2010 ; nich10 . Therefore, decoherence-free subspaces
Lidar1998 ; Albert2016 ; Tarnowski2021  (DFS), which are protected against
decoherence effects Habib1998 , play a crucial role in realizing quantum
computing Grover1997 ; Mohseni2003 ; Friesen2017 . Understanding and
formulating the general conditions underlying DFS is the prerequisite to tame
decoherence and to establish open systems which exhibit non-stationary long-
time dynamics (NLD), that is, remain out of equilibrium despite their
interaction with an environment Buca2019 ; Tindall2020 ; Tarnowski2021 .

In closed systems, it is well known that symmetries, i.e., constants of
motion, are crucial to characterize their dynamics. Here, we will show that
this is also the case for open systems: Symmetries can be used to formulate
simple conditions for DFS in terms of the dependence of the system and the
coupling to the environment on these symmetries. Moreover, these conditions
can be identified and formulated in an intuitive way via classical dynamics
since exact symmetries hold quantum mechanically as well as classically
through their equivalent formulation in terms of commutators and Poisson
brackets, respectively. The classical phase-space perspective will also allow
us to introduce naturally new classes of DFS where the coupling to the
environment depends only on so called “ghost variables”, orthogonal to the
symmetries. Finally, we will present the quantized conditions for the DFS in
terms of eigenstates of the symmetry operator.

Our starting point is the Lindblad master equation Breuer2002
$\dot{\rho}{=}-\mathrm{i}\mathcal{L}(\mathbb{H},\mathbb{L})\rho$ for the
systems’s density matrix $\rho$ with Hamiltonian $\mathbb{H}$ coupled to the
the environment in Markov approximation through the Lindblad operators
$\mathbb{L}$. In the framework of the master equation DFS exist if and only if
the Lindbladian $\mathcal{L}$ has nonzero real eigenvalues Lidar1998 ;
Baumgartner2008 ; Albert2014 ; Buca2019 . Here, we will work with the adjoint
Lindbladian $\mathcal{L}^{\dagger}(\mathbb{H},\mathbb{L})$ which has identical
spectral properties, since we are interested in operators, most prominently a
symmetry $\mathbb{J}$. As a constant of motion for the open system its
dynamics is governed in the Heisenberg picture by $\mathcal{L}^{\dagger}$
through

$\displaystyle 0=$ $\displaystyle\frac{{\rm d}\mathbb{J}}{{\rm
d}t}=\mathrm{i}\mathcal{L}^{\dagger}\mathbb{J},$ (1) with
$\mathcal{L}^{\dagger}=\mathcal{L}^{\dagger}_{\mathbb{H}}+\mathcal{L}^{\dagger}_{\mathbb{L}}$,
where $\displaystyle\mathcal{L}^{\dagger}_{\mathbb{H}}\mathbb{J}=$
$\displaystyle\dfrac{1}{\hbar}[\mathbb{H},\mathbb{J}],$ (1a)
$\displaystyle\mathcal{L}^{\dagger}_{\mathbb{L}}\mathbb{J}=$
$\displaystyle\dfrac{\mathrm{i}}{\hbar}\sum_{\alpha}\left(\mathbb{L}^{\dagger}_{\alpha}[\mathbb{J},\mathbb{L}_{\alpha}]+[\mathbb{L}^{\dagger}_{\alpha},\mathbb{J}]\mathbb{L}_{\alpha}\right)\,.$
(1b)

Since symmetries are respected equivalently by a quantum system and its
classical counterpart (here, via Poisson brackets), we can give an intuitive
account of these conditions in classical phase space Strunz1998 ; Dubois2021 .
For a constant of motion $J$ with
$\\{H,J\\}\,{=}\,\\{L_{\alpha},J\\}\,{=}\,0$, we find that semiclassically DFS
exist if there is at least one layer $J{=}J_{0}$ on which

$\omega(J_{0})\equiv\left.\partial H/\partial J\right|_{J_{0}}\neq
0,\quad\ell_{\alpha}(J_{0})\equiv\left.\partial L_{\alpha}/\partial
J\right|_{J_{0}}=0,$ (2)

where $\omega(J)$ is the Hamiltonian frequency and $\ell_{\alpha}(J)$ can be
interpreted as the decay rate induced by the environment. Note, that we denote
quantum operators with $\mathbb{X}$ in contrast to (classical) functions $X$.
While the condition on $H$ is necessary for (oscillatory) non-stationary
dynamics in the first place, the condition on $L_{\alpha}$ makes sure that
this dynamics is preserved for long times and therefore establishes NLD.
Interestingly, since $L_{\alpha}{=}L_{\alpha}(J,\mathbf{Z})$ is in principle a
function of all phase-space variables (boldface letters denote a set of
variables), the condition $\ell_{\alpha}(J){=}0$ can be achieved in two,
qualitatively different ways:

  * (i)

local realization for a specific $J_{0}$: the $L_{\alpha}$ depend explicitly
on $J$ but their derivative vanishes for $J_{0}$;

  * (ii)

global realization for all $J$: the $L_{\alpha}$ do not depend on $J$ but can
depend on all the variables $\mathbf{Z}$, the _ghost variables_ , being
orthogonal to $J$.

These two ways exist also quantum mechanically, where the conditions on the
symmetry (2) are quantized in the form that there exist two eigenspaces
$\\{|n\rangle\\}$ and $\\{|m\rangle\\}$ of $\mathbb{J}$ with eigenvalues
$J_{n}$ and $J_{m}$, respectively, such that

$\Delta\mathbb{H}_{nm}\neq 0\quad\mbox{and}\quad\Delta\mathbb{L}_{\alpha
nm}=0,$ (3)

with the explicit forms of $\Delta\mathbb{H}_{\alpha nm}$ and
$\Delta\mathbb{L}_{\alpha nm}$ given below in Eq. (10).

Figure 1: Autocorrelation function ${\rm tr}\rho(t)\rho(t^{\prime})$ for
Hamiltonian (6) with $N{=}6$. Parameters $\omega_{i}$, $\delta_{ij}$ and
$\Delta_{ij}$ are given in the supplement SuppMat . (a) No dissipation,
$\mathcal{L}_{\mathbb{L}}=0$. (b) Dissipation with $N$ Lindblad operators
$\mathbb{L}_{\alpha}=\gamma\mathbb{S}_{\alpha}^{z}$. (c) Dissipation with
$(N{-}1)N/2$ ghost operators such that
$\mathbb{L}_{\alpha_{jk}}=\gamma(\mathbb{S}_{j}^{z}-\mathbb{S}_{k}^{z})$. In
both cases $\gamma{=}1/3$.

Since systems which we classify now as local realizations (i) have been
discussed in Buca2019 ; Tindall2020 , we focus on global realizations (ii) in
the following. Figure 1 shows such a global realization with a Heisenberg spin
model in comparison to purely Hamiltonian dynamics (Fig. 1a) and to the case
with an arbitrary dissipative part (Fig. 1b). Indeed, for an environment
depending on ghost operators only (Fig. 1c), coherent oscillations are
preserved despite dissipation. Before we construct the ghost operators
explicitly via the corresponding classical ghost variables, we derive the
conditions (2).

Consider phase-space variables $(J,\theta,\mathbf{Z})$ with $J$ conserved,
$\\{J,H\\}\,{=}\,\\{J,L_{\alpha}\\}\,{=}\,0$. The angle $\theta$ is
canonically conjugate to the action-like $J$ such that
$\\{\theta,J\\}\,{=}\,1$. Then, the phase space is foliated by manifolds on
which $J$ is conserved. The $\mathbf{Z}$ are a set of canonically (or
noncanonically) conjugate variables such that
$\\{\mathbf{Z},J\\}\,{=}\,\\{\mathbf{Z},\theta\\}\,{=}\,\boldsymbol{0}$. The
relevance of the _ghost variables_ $\mathbf{Z}$ for DFS will become clear
below.

From the action-angle variables we can construct “beating variables”

$A=f(J)\exp(-\mathrm{i}\theta)\,\,\,\,\text{and}\,\,\,\,A^{*}=f(J)\exp(\mathrm{i}\theta)\,,$
(4)

with $f(J)=A^{*}A$ Campoamor2012 ; SuppMat . The beating variables describe
dynamics along a path on which $J$ is conserved. In the semiclassical limit of
the adjoint Lindbladian Strunz1998 ; Dubois2021 ; SuppMat  the dynamics of $A$
on the manifold of constant $J$ is given by
$\dot{A}=\mathrm{i}\mathcal{L}^{*}A$ with

$\mathcal{L}^{*}A={-}\bigg{(}\partial_{J}H+\sum_{\alpha}\Big{(}2{\rm
Im}\,(L_{\alpha}\partial_{J}L_{\alpha}^{*})-\mathrm{i}\hbar|\partial_{J}L_{\alpha}|^{2}\Big{)}\bigg{)}\,A,$
(5)

which follows from the Poisson brackets
$\\{A,H\\}\,{=}\,{-}\mathrm{i}(\partial_{J}H)\,A$ and
$\\{A,L_{\alpha}\\}\,{=}\,{-}\mathrm{i}(\partial_{J}L_{\alpha})\,A$, where
$\partial_{J}H$ is real and $\partial_{J}L_{\alpha}$ can be complex (since
$L_{\alpha}$ can be complex). Hence, $A$ evolves in time with an oscillatory
and a decaying part, regardless of the dynamics of the ghost variables
$\mathbf{Z}$. The oscillatory part comes from a combination of the Hamiltonian
and the dissipative components, while the decay part is a consequence of
dissipation only. Note, that the decaying part is of order $\hbar$ and is a
consequence of diffusion or quantum noise.

If $\partial_{J}L_{\alpha}{=}0$ the decaying part in (5) vanishes and the
beating variable $A$ exhibits NLD, provided that $\partial_{J}H\neq 0$ which
leads to the conditions (2) specifying the existence of DFS. If the Lindblad
function depends explicitly on the conserved quantity $J$ and there exists a
manifold $J{=}J_{0}$ on which $\ell_{\alpha}(J_{0},\mathbf{Z}){=}0$, the
environment couples to the degree of freedom associated with $J$, but not on
the subspace $J{=}J_{0}$. Although $J$ is conserved, NLD is restricted to the
manifold $J{=}J_{0}$, and therefore we call this realization (i) of DFS
_local_. One also gets $\ell_{\alpha}(J,\mathbf{Z}){=}0$, if $L_{\alpha}$ does
not depend explicitly on $J$ at all. In this case, the environment does not
couple to the degree of freedom associated with $J$. These conditions apply to
all manifolds labeled by $J$, and therefore this constitutes a _global_
realization (ii) of DFS.

In both cases, the environment does not affect the oscillatory dynamics of the
beating variable $A$ characterized by the (non-zero) real eigenvalues
$\omega(J,\mathbf{Z})$ of the Lindbladian. If the Hamiltonian frequency
$\omega$ depends explicitly on the ghost variables $\mathbf{Z}$, this
oscillatory dynamics can be very complicated. If $\omega{=}\omega(J)$ only,
one can directly solve $\dot{A}=\mathrm{i}\mathcal{L}^{*}A$ to obtain
$A(t)=\exp(-\mathrm{i}\omega(J)t)A(0)$. In case (i) the environment
extinguishes for long times all oscillatory motion linked to $J$ but one with
frequency $\omega(J_{0})$ on the manifold $J_{0}$.

From the semiclassical perspective as developed above we can draw further
conclusions. The conditions (2) do not require integrability (or near-
integrability) of the system. Therefore, also classically chaotic systems can
have DFS, which we demonstrate here explicitly with the Heisenberg spin model.
Furthermore, if DFS exist due to a symmetry, small perturbations of the system
will not destroy them. This follows from the KAM theorem delaLlave2005 , which
ensures the existence of modified action-angle variables for open systems
subjected to perturbations. Therefore, we construct now explicitly a coupling
to the environment according to (ii) for a chaotic Hamiltonian.

#### Application to the Heisenberg XXZ spin model:

Figure 2: (a) Schematic of the spin system for $N{=}6$ governed by Hamiltonian
(6) as used before in Fig. 1. The red lines illustrate the dissipations
induced by the ghost operators. (b) and (c) Spectrum of the adjoint of the
Lindbladian $\mathcal{L}^{\dagger}$. The cyan circles are the purely real
eigenvalues. (b) For $(N{-}1)N/2$ Lindblad operators as a linear combination
of the ghost operators (8a) such that
$\mathbb{L}_{\alpha_{ij}}=\gamma(\mathbb{S}_{i}^{z}-\mathbb{S}_{j}^{z})$. (c)
For $N{-}1$ Lindblad operators as a function of the ghosts (8b) with
$\mathbb{L}_{\alpha}=\gamma\widetilde{\mathbb{S}}_{\alpha}^{+}$. In both cases
$\gamma{=}1/3$.

A realization of (ii) requires the Lindblad operators $\\{\mathbb{Z}_{i}\\}$
to be independent of the symmetry $\mathbb{J}$ of the system. Therefore, only
many-body systems render this case nontrivial. We consider the Heisenberg XXZ
spin model Prosen2011 ; Ganahl2012 ; Liu2014 ; Sa2020

$\mathbb{H}=\sum_{i}\omega_{i}\mathbb{S}^{z}_{i}+\sum_{ij}\left[\sigma_{ij}\left(\mathbb{S}^{+}_{i}\mathbb{S}^{-}_{j}{+}\mathbb{S}^{-}_{i}\mathbb{S}^{+}_{j}\right)+\Delta_{ij}\mathbb{S}^{z}_{i}\mathbb{S}^{z}_{j}\right],$
(6)

where the
$\mathbf{\mathbb{S}}_{i}{=}(\mathbb{S}^{x}_{i},\mathbb{S}^{y}_{i},\mathbb{S}^{z}_{i})$
are spin $1/2$ operators on site $i$ from Pauli matrices, and
$\mathbb{S}^{\pm}_{i}=\mathbb{S}^{x}_{i}\pm\mathrm{i}\,\mathbb{S}^{y}_{i}$ are
ladder operators, with $i=1,...,N$ and $N$ the number of spins, evoking spin
flips between site $i$ and $j$ in the combinations
$\mathbb{S}^{\pm}_{i}\mathbb{S}^{\mp}_{j}$. The total spin
$\mathbb{J}=\mathbb{S}^{z}=\sum_{k{=}1}^{N}\mathbb{S}^{z}_{k}/N$ is conserved
in time as $\mathcal{L}^{\dagger}\mathbb{J}{=}0$ and this leads to pertinent
oscillations associated with $\mathbb{J}$, see Fig. 1c. The parameters used in
Eq. (6) are given in the supplemental material SuppMat .

We construct now an environment depending on ghost operators
$\\{\mathbb{Z}_{i}\\}$ only which does not affect these oscillations. First,
we derive classical ghost variables with canonical transformations and
transfer the ghost variables to ghost operators in the quantum domain. To this
end, we introduce polar coordinates
$\mathbf{S}_{i}=[S_{i}^{2}-(S^{z}_{i})^{2}]^{1/2}(\mathbf{e}_{x}\cos\theta_{i}+\mathbf{e}_{y}\sin\theta_{i})+\mathbf{e}_{z}S^{z}_{i}$
for the spin on site $i$, where the azimuthal angles $\theta_{i}$ are
canonically conjugate to the $S^{z}_{i}$ and
$S_{i}{}^{2}{=}|\mathbf{S}_{i}|^{2}$. Obviously, Hamiltonian (6) is invariant
under rotation with arbitrary azimuthal angle and therefore
$J=S^{z}/N=\sum_{i}S^{z}_{i}/N$ is conserved. Akin to relative center-of-mass
motion for massive particles we take advantage of this fact by introducing
relative angles $\Delta\theta_{k}=\theta_{k{+}1}-\theta_{k}$ for
$k=1,...,N-1$, and the total angle $\theta=\sum_{i}\theta_{i}$. They are
related to the Jacobi coordinates for celestial many-body systems Cabral2002 .

The canonical transformation from phase-space variables
$(\theta_{i},S^{z}_{i})$ to phase-space variables $(J,\theta,\mathbf{Z})$,
with ghost variables $\mathbf{Z}=(\Delta\theta_{k},\widetilde{S}_{k})$ for
$k=1,...,N-1$, and its inverse are given in the supplemental material SuppMat
. The transformation is obtained by using the canonical rules Goldstein1980
with a suitable generating function associated with the center of mass and the
relative angles. Besides $\Delta\theta_{k}$, we obtain as ghost variables
SuppMat

$\widetilde{S}_{k}^{z}=\dfrac{k}{N}\sum_{i{=}1}^{N}S^{z}_{i}-\sum_{i{=}1}^{k}S^{z}_{i},$
(7)

which are linear functions of the spin variables. Note that using canonical
transformations ensures that the new set of variables spans the entire phase
space. If the interaction with the environment is any combination of these
variables or operators (i.e., if it does not depend explicitly on $\theta$ and
$J$) DFS exist and the open system exhibits NLD. The ghost variables or
operators do not affect the oscillatory dynamics associated with the conserved
quantity $J$. Therefore, coherent oscillations persist on long-timescales and
the spectrum of the Lindbladian contains nonzero real eigenvalues, cf. Fig. 2.

The beating variables associated with the conserved quantity, as given in Eq.
(4), are obtained in terms of spin variables by performing a non-canonical
change of coordinates SuppMat  leading to $A\propto\prod_{i{=}1}^{N}S^{-}_{i}$
and $A^{*}\propto\prod_{i{=}1}^{N}S^{+}_{i}$. The beating variables oscillate
at a frequency $\Omega=\sum_{i{=}1}^{N}\omega_{i}$, corresponding to the
absolute value of the nonzero real eigenvalues and are not affected by
dissipation. The _ghost operators_ as the quantum analogs of the _ghost
variables_ follow simply from replacing the Cartesian spin variables in the
ghost variable expressions by the corresponding Cartesian spin operators.
Therefore, the ghost operators are functions of the spin operators (Pauli
matrices) and read

$\displaystyle\widetilde{\mathbb{S}}_{k}^{z}=\dfrac{k}{N}\sum_{i{=}1}^{N}\mathbb{S}^{z}_{i}-\sum_{i{=}1}^{k}\mathbb{S}^{z}_{i},$
(8a)
$\displaystyle\widetilde{\mathbb{S}}_{k}^{\pm}=\mathbb{D}_{k}\;\mathbb{S}^{\pm}_{k+1}\,\mathbb{S}^{\mp}_{k},$
(8b)

with $k=1,...,N{-}1$ and ia diagonal matrix $\mathbb{D}_{k}$ SuppMat . By
construction, all combinations of ghost operators are also ghost operators.
Any Lindbladian which is an analytic function of these ghost operators leads
to DFS.

In Fig. 2b, we have used $(N{-}1)N/2$ Lindblad operators as a linear
combination of the ghost operators (8a) such that
$\mathbb{L}_{\alpha_{jk}}=\gamma(\mathbb{S}_{j}^{z}-\mathbb{S}_{k}^{z})$ with
$\gamma{=}1/3$ and $\alpha_{ij}=(j{-}1)(j{-}2)/2{+}i$ for $j=2,...,N$ and
$i=1,...,j{-}1$. For Fig. 2c, we have used $N{-}1$ Lindblad operators of the
ghost operators (8b) such that
$\mathbb{L}_{\alpha}=\gamma\widetilde{\mathbb{S}}_{\alpha}^{+}$ with
$\gamma{=}1/3$ and $\alpha=1,...,N-1$. In both cases, the coherent
oscillations associated with the symmetry $\mathbb{J}$ are not affected,
neither are the beating operators
$\mathbb{A}=\prod_{i{=}1}^{N}\mathbb{S}^{-}_{i}$ and
$\mathbb{A}^{\dagger}=\prod_{i{=}1}^{N}\mathbb{S}^{+}_{i}$. Note also, that
the (real) eigenvalues (cyan circles) are the same for the cases of Fig. 2b
and 2c, since they come from the Hamiltonian part of the Lindbladian
$\mathcal{L}_{\mathbb{H}}^{\dagger}$ which we have not changed. We find
$\mathcal{L}^{\dagger}_{\mathbb{H}}\mathbb{A}{=}\Omega\mathbb{A}$ and
$\mathcal{L}^{\dagger}_{\mathbb{L}}\mathbb{A}{=}0$ corresponding to the cyan
circles for positive real eigenvalues (the negative one corresponds to the one
associated with $\mathbb{A}^{\dagger}$) in Figs. 2b and 2c.

#### Theory of symmetry-induced DFS in quantum mechanics:

Finally, we are in a position to briefly sketch the derivation of the
quantized condition (3) for DFS. We have a set of commuting operators,
$\\{\mathbb{Z}_{i}\\}$, its conjugate $\mathbb{J}$ and and the ghost
operators. Together, they span the Hilbert space for the open system dynamics.

The Hamilton operator $\mathbb{H}$ (and similarly all Lindblad operators
$\mathbb{L}_{\alpha}$) with the symmetry $\mathbb{J}$ can be written as a
direct sum of irreducible (square) matrices

$\mathbb{H}=\bigoplus_{n}\mathbb{H}_{n},\qquad\mathbb{L}_{\alpha}=\bigoplus_{n}\mathbb{L}_{\alpha
n},$ (9)

where $\mathbb{X}_{n}$ is associated with the eigenspace of $J_{n}$, an
eigenvalue of $\mathbb{J}$. They are defined as $\mathbb{X}_{n}\,{=}\,{\rm
tr}_{n}\mathbb{P}_{n}\mathbb{X}\mathbb{P}_{n}$ where
$\mathbb{P}_{n}\,{=}\,\prod_{j(J_{\\!j}\neq
J_{n})}(\mathbb{J}{-}J_{\\!j}\mathbb{1})/(J_{n}{-}J_{\\!j})$ are projectors
Lowdin1962  rendering all subspaces to null operators but the one associated
with the eigenvalue $J_{n}$ of $\mathbb{J}$. ${\rm tr}_{n}$, finally, takes
the partial trace over all complementary subspaces, reducing the dimension of
$\mathbb{X}_{n}$ to the irreducible one of $\mathbb{J}_{n}$. Corresponding to
the multiplicity $\eta_{n}$ of the eigenvalue $J_{n}$, the $\mathbb{H}_{n}$
and $\mathbb{L}_{\alpha n}$ are $\eta_{n}{\times}\eta_{n}$ matrices. Due to
the block-diagonal form (9), the superoperator $\mathcal{L}^{\dagger}$ can be
written as blocks associated with a pair of eigenvalues $(J_{n},J_{m})$ such
that $\mathcal{L}^{\dagger}{=}\bigoplus_{n,m}\mathcal{L}_{nm}^{\dagger}$. Each
block of the superoperator

${\cal
L}^{\dagger}_{nm}=\frac{1}{\hbar}\Delta\mathbb{H}_{nm}+\frac{\mathrm{i}}{\hbar}\sum_{\alpha}\big{(}\mathbb{L}^{\dagger}_{\alpha
n}\Delta\mathbb{L}_{\alpha nm}+\Delta\mathbb{L}^{\dagger}_{\alpha
nm}\mathbb{L}_{\alpha m}\big{)}$ (10) is of size
$(\eta_{n}{\times}\eta_{m}){\times}(\eta_{n}{\times}\eta_{m})$ with Eins
$\displaystyle\Delta\mathbb{H}_{nm}$
$\displaystyle\equiv\mathbb{H}_{n}\otimes\mathbb{1}_{m}-\mathbb{1}_{n}\otimes\mathbb{H}_{m}^{\top},$
(10a) $\displaystyle\Delta\mathbb{L}_{\alpha nm}$
$\displaystyle\equiv\mathbb{L}_{\alpha
n}\otimes\mathbb{1}_{m}-\mathbb{1}_{n}\otimes\mathbb{L}_{\alpha m}^{\top}.$
(10b)

Coherent oscillations of a beating operator $\mathbb{A}_{nm}$, which is an
eigenoperator of ${\cal L}^{\dagger}_{nm}$ with nonzero real eigenvalues,
occur for those $\Delta\mathbb{H}_{nm}$ and $\Delta\mathbb{L}_{\alpha nm}$
that fulfill condition (3). The eigenoperators $\mathbb{A}_{nm}$ are matrices
with an $\eta_{n}{\times}\eta_{m}$ non-vanishing part such that
$\mathbb{A}_{nm}\,{=}\,\mathbb{P}_{n}\mathbb{A}_{nm}\mathbb{P}_{m}$ and can
have multiple beating frequencies. Furthermore, the $\mathbb{A}_{nm}$ are also
eigenoperators of the Lindbladian since under condition (3), with (10b),
$[\mathbb{L}_{\alpha n},\mathbb{L}_{\alpha
n}^{\dagger}]\,{=}\,[\mathbb{L}_{\alpha m},\mathbb{L}_{\alpha
m}^{\dagger}]\,{=}\,0$. The fact that the Lindbladian and its adjoint share
the same eigenoperators associated to real eigenvalues show that dissipation
plays a passive role in this case.

This completes our classical, semiclassical and quantum treatment of symmetry-
induced DFS. It clearly reveals that the notion of non-stationary
(oscillating) behavior and its equivalence to real eigenvalues of the
Lindbladian applies equally to classical, semiclassical and quantum systems
which establishes DFS for classical and semiclassical systems. For the latter,
while coherent oscillations in classical systems can still occur in the
presence of dissipation (through the second term in (5), see Keeling2010 ;
Bhaseen2012 ; Munoz2019  for examples), diffusion can inhibit long-time
oscillatory non-stationary motion (through the third term in (5) being of
order $\hbar$). The eigenoperator of the adjoint Lindbladian corresponding to
the real nonzero eigenvalues can be constructed classically, see Eq. (4), or
quantum mechanically ($\mathbb{A}_{nm}$) from a symmetry. Playing the role of
a beating operator, $A$ or $\mathbb{A}_{mn}$ connect different elements of the
constant of motion $J$ or $\mathbb{J}$ in form of its layers in phase space or
its irreducible representations in Hilbert space, respectively. Elements
$\mathbb{J}_{n}$ of the symmetry itself can be expressed in terms of the
beating operator as
$\mathbb{J}_{n}\,{=}\,\mathbb{A}_{mn}^{\dagger}\mathbb{A}_{mn}$. This
irreducible representation also plays a crucial role in the context of
breaking the ergodicity thermalization hypothesis in Hamiltonian systems
Sala2020 , which is related to NLD in the absence of dissipation. For DFS, the
Lindblad operators must be degenerate with respect to at least two different
irreducible subspaces $n\neq m$ of the symmetry, while the hamiltonian must
not be degenerate, see (3). This gives a clear picture how DFS emerge.

Guided by this insight, we have identified a new class of couplings to the
environment leading to DFS, namely when the Lindblad operators do only depend
on ghost operators which span the Hilbert space but commute with the symmetry
$\mathbb{J}$, as explicitly demonstrated with the Heisenberg XXZ spin model.
In terms of the eigenoperator, DFS exist if
$[\mathbb{L}_{\alpha},\mathbb{A}_{nm}]\,{=}\,[\mathbb{L}_{\alpha}^{\dagger},\mathbb{A}_{nm}]\,{=}\,0$
Buca2019 . Note, however, that the latter condition is not necessary since for
the adjoint Lindbladian to have real eigenvalues, it follows from (1b) that
only the sum
$\mathbb{L}_{\alpha}^{\dagger}[\mathbb{A},\mathbb{L}_{\alpha}]{+}[\mathbb{L}^{\dagger}_{\alpha},\mathbb{A}]\mathbb{L}_{\alpha}\,{=}\,0$
must vanish. How this extends the classes of possible interactions with the
environment leading to DFS even further beyond those depending only on ghost
operators will be a subject of further research in_preparation .

#### Acknowledgments:

JD acknowledges Andreas Buchleitner, Charlie Duclut, Christian Johansen and
Francesco Piazza for helpful discussions.

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Symmetry-induced decoherence-free subspaces
– Supplemental material –
Jonathan Dubois, Ulf Saalmann and Jan Michael Rost

##  I Lindbladian operator, its adjoint operator and its semiclassical limit

In the quantum mechanical context, the inner product between two observables
$\mathbb{F}$ and $\mathbb{G}$ in the Hilbert space is given by
$\langle\mathbb{F},\mathbb{G}\rangle={\rm
tr}\;(\mathbb{F}^{\dagger}\mathbb{G})$ where $\mathbb{F}^{\dagger}$ denotes
the complex-conjugate transpose of $\mathbb{F}$. The adjoint
$\mathcal{L}^{\dagger}$ of the Lindbladian is defined through
$\langle\mathcal{L}^{\dagger}\mathbb{F},\mathbb{G}\rangle=\langle\mathbb{F},\mathcal{L}\mathbb{G}\rangle$.
The Lindbladian and its adjoint are given by Albert2014

$\displaystyle\mathcal{L}\rho$
$\displaystyle=\dfrac{1}{\hbar}[\mathbb{H},\rho]+\dfrac{{\rm
i}}{\hbar}\sum_{\alpha}\left([\mathbb{L}_{\alpha},\rho\mathbb{L}^{\dagger}_{\alpha}]+[\mathbb{L}_{\alpha}\rho,\mathbb{L}_{\alpha}^{\dagger}]\right),$
(S11a) $\displaystyle\mathcal{L}^{\dagger}\mathbb{F}$
$\displaystyle=\dfrac{1}{\hbar}[\mathbb{H},\mathbb{F}]-\dfrac{{\rm
i}}{\hbar}\sum_{\alpha}\left(\mathbb{L}_{\alpha}^{\dagger}[\mathbb{F},\mathbb{L}_{\alpha}]+[\mathbb{L}^{\dagger}_{\alpha},\mathbb{F}]\mathbb{L}_{\alpha}\right),$
(S11b)

respectively, such that we have $\dot{\rho}=-{\rm i}\mathcal{L}\rho$ and
$\dot{\mathbb{F}}={\rm i}\mathcal{L}^{\dagger}\mathbb{F}$. In the limit
$\mathbb{L}_{\alpha}{=}0$ it is $\mathcal{L}{=}\mathcal{L}^{\dagger}$. In the
semiclassical limit, the scalar product in the Hilbert space with phase-space
variables $\mathbf{z}$ is given by $\langle F,G\rangle=\int\\!{\rm
d}\rule{-0.50003pt}{8.1805pt}^{n}\\!\\!z\,F^{*}(\mathbf{z})G(\mathbf{z})$ for
an $n$-dimensional phase space. The semiclassical limit of the Lindbladian and
its adjoint are given by Dubois2021

$\displaystyle\mathcal{L}\rho$ $\displaystyle={\rm
i}\\{H,\rho\\}-\sum_{\alpha}\Big{(}\left(\\{L_{\alpha},\rho
L_{\alpha}^{*}\\}+\\{L_{\alpha}\rho,L_{\alpha}^{*}\\}\right)-\dfrac{{\rm
i}\hbar}{2}\left(\\{\\{L_{\alpha},\rho\\},L_{\alpha}^{*}\\}+\\{L_{\alpha},\\{\rho,L_{\alpha}^{*}\\}\\}\right)\Big{)},$
(S12a) $\displaystyle\mathcal{L}^{*}F$ $\displaystyle={\rm
i}\\{H,F\\}+\sum_{\alpha}\Big{(}\left(L_{\alpha}^{*}\\{F,L_{\alpha}\\}+\\{L_{\alpha}^{*},F\\}L_{\alpha}\right)+\dfrac{{\rm
i}\hbar}{2}\left(\\{\\{L_{\alpha},F\\},L_{\alpha}^{*}\\}+\\{L_{\alpha},\\{F,L_{\alpha}^{*}\\}\\}\right)\Big{)}.$
(S12b)

The Lindbladian can always be cast into a Fokker-Planck equation Dubois2021 ;
Strunz1998 .

##  II Heisenberg spin model

We provide details of the classical construction of ghost variables in order
to define environments that allow for decoherence-free subspaces for the
Heisenberg model presented in the text.

### Classical spin algebra

We consider a chain of $N$ spins with in the classical limit is described by
the spin variables
$\mathbf{S}_{i}=S^{x}_{i}\mathbf{e}_{x}+S^{y}_{i}\mathbf{e}_{y}+S^{z}_{i}\mathbf{e}_{z}$
with $i=1,...,N$ and the spin algebra

$\\{S^{\alpha}_{i},S^{\beta}_{j}\\}=\delta_{ij}\varepsilon_{\alpha\beta\gamma}S^{\gamma}_{i}$
(S13)

with $\varepsilon_{\alpha\beta\gamma}$ the Levi-Civita symbol. The
corresponding non-canonical Poisson bracket is given by

$\\{F,G\\}=\sum_{i=1}^{N}\mathbf{S}_{i}\cdot\dfrac{\partial
F}{\partial\mathbf{S}_{i}}\times\dfrac{\partial G}{\partial\mathbf{S}_{i}}.$
(S14a) Equivalently, we can describe the dynamics with the beating variables
$S^{\pm}_{i}=S^{x}_{i}\pm{\rm i}S^{y}_{i}$, for which the Poisson bracket
reads $\displaystyle\\{F,G\\}={}$ $\displaystyle\sum_{i=1}^{N}\bigg{(}{-}2{\rm
i}S^{z}_{i}\bigg{(}\dfrac{\partial F}{\partial S^{+}_{i}}\dfrac{\partial
G}{\partial S^{-}_{i}}-\dfrac{\partial F}{\partial S^{-}_{i}}\dfrac{\partial
G}{\partial S^{+}_{i}}\bigg{)}$ $\displaystyle\qquad+{\rm
i}\,S^{+}_{i}\bigg{(}\dfrac{\partial F}{\partial S^{+}_{i}}\dfrac{\partial
G}{\partial S^{z}_{i}}-\dfrac{\partial F}{\partial S^{z}_{i}}\dfrac{\partial
G}{\partial S^{+}_{i}}\bigg{)}-{\rm i}\,S^{-}_{i}\bigg{(}\dfrac{\partial
F}{\partial S^{-}_{i}}\dfrac{\partial G}{\partial S^{z}_{i}}-\dfrac{\partial
F}{\partial S^{z}_{i}}\dfrac{\partial G}{\partial S^{-}_{i}}\bigg{)}\bigg{)},$
(S14b)

with the fundamental spin algebra $\\{S^{z}_{i},S^{\pm}_{i}\\}=\mp{\rm
i}S^{\pm}_{i}$ and $\\{S^{+}_{i},S^{-}_{i}\\}=-2{\rm i}S^{z}_{i}$. Note that
in both cases, the classical Poisson bracket is non-canonical and is related
to the quantum Lie bracket by $[\mathbb{F},\mathbb{G}]\equiv{\rm
i}\hbar\\{F,G\\}$.

### Cylindrical coordinates and conserved quantity

In cylindrical coordinates $\\{S_{i}^{z},\theta_{i}\\}$ it is
$\mathbf{S}_{i}=\sqrt{S_{i}{}^{2}{-}(S^{z}_{i})^{2}}\,(\mathbf{e}_{x}\cos\theta_{i}+\mathbf{e}_{y}\sin\theta_{i})+S^{z}_{i}\mathbf{e}_{z}$
or equivalently
$S^{\pm}_{i}=\sqrt{S_{i}{}^{2}{-}(S_{i}^{z})^{2}}\,\exp(\pm{\rm i}\theta)$
with $S_{i}{}^{2}=|\mathbf{S}_{i}|^{2}$ a Casimir invariant. Therewith the
Hamiltonian (in the semiclassical limit) reads

$H(\theta_{i},S^{z}_{i})=\sum_{i=1}^{N}\bigg{(}\omega_{i}S^{z}_{i}+\sum_{j=1}^{N}\left(2\sigma_{ij}\sqrt{[S_{i}^{2}-(S^{z}_{i})^{2}][S_{j}^{2}-(S^{z}_{j})^{2}]}\cos\left(\theta_{i}-\theta_{j}\right)+\Delta_{ij}S^{z}_{i}S^{z}_{j}\right)\bigg{)},$
(S15)

and the Poisson bracket becomes

$\\{F,G\\}=\sum_{i=1}^{N}\bigg{(}\dfrac{\partial
F}{\partial\\!\theta_{i}}\dfrac{\partial G}{\partial
S^{z}_{i}}-\dfrac{\partial F}{\partial S^{z}_{i}}\dfrac{\partial
G}{\partial\\!\theta_{i}}\bigg{)}.$ (S16)

From the Hamiltonian (S15), it is clear that

$J=\dfrac{1}{N}\sum_{i=1}^{N}S^{z}_{i},$ (S17)

is a conserved quantity due to the invariance under a rotation around
$\mathbf{e}_{z}$. The variable canonically conjugate to $J$ such that
$\\{\theta,J\\}=1$ is given by

$\theta=\sum_{i=1}^{N}\theta_{i},$ (S18)

corresponding to the total angle. There is some freedom for choosing the set
of canonically-conjugate variables from there and therefore the _ghost
variables_. Indeed, note that one can transform one set of ghost variables
into another one by canonical transformations. The most natural starting point
are the Jacobi coordinates for the celestial many-body problem
$\Delta\theta_{i}=\theta_{i+1}-\theta_{i}$. It is clear that
$\\{\Delta\theta_{i},J\\}=\\{\Delta\theta_{i},\theta\\}=0$ for $i=1,...,N-1$.
The goal is now to find the set of variables $\widetilde{S}^{z}_{i}$
canonically conjugate to $\Delta\theta_{i}$, i.e.,
$\\{\Delta\theta_{i},\widetilde{S}^{z}_{j}\\}=\delta_{ij}$ with $\delta_{ij}$
the Kronecker delta.

### Ghost variables using canonical transformations

For doing so, we use the $F_{2}$ generating function Goldstein1980 . We recall
that for a set of canonically conjugate variables
$(\boldsymbol{\theta},\mathbf{S}^{z})$ the new set of canonically conjugate
variables $(\widetilde{\boldsymbol{\theta}},\widetilde{\mathbf{S}}^{z})$ can
be found by using transformations, such that
$F_{2}(\boldsymbol{\theta},\widetilde{\mathbf{S}}^{z})$, such that

$\mathbf{S}^{z}=\dfrac{\partial
F_{2}}{\partial\boldsymbol{\theta}},\qquad\widetilde{\boldsymbol{\theta}}=\dfrac{\partial
F_{2}}{\partial\widetilde{\mathbf{S}}^{z}}.$ (S19)

Here, the old variables are given by
$\boldsymbol{\theta}=(\theta_{1},...,\theta_{N})$ and
$\mathbf{S}^{z}=(S^{z}_{1},...,S^{z}_{N})$. The new variables are
$\widetilde{\boldsymbol{\theta}}=(\Delta\theta_{1},...,\Delta\theta_{N{-}1},\theta)$
and
$\widetilde{\mathbf{S}}^{z}=(\widetilde{S}^{z}_{1},...,\widetilde{S}^{z}_{N{-}1},J)$.
Given the form of the new variables $\widetilde{\boldsymbol{\theta}}$ which we
have imposed, the generating function is given by

$F_{2}(\boldsymbol{\theta},\widetilde{\mathbf{S}}^{z})=J\sum_{i=1}^{N}\theta_{i}+\sum_{i=1}^{N{-}1}\widetilde{S}^{z}_{i}(\theta_{i+1}-\theta_{i}).$
(S20)

Indeed, from (S19) and (S20), we obtain that
$\widetilde{\theta}_{0}\equiv\theta=\sum_{i=1}^{N}\theta_{i}$ and
$\widetilde{\theta}_{i}\equiv\Delta\theta_{i}=\theta_{i+1}-\theta_{i}$ for
$i=1,...,N{-}1$. The expression of the old momenta with respect to the new
ones are found using (S19) as

$S^{z}_{1}=J-\widetilde{S}^{z}_{1},\qquad
S^{z}_{i}=J+\widetilde{S}^{z}_{i{-}1}-\widetilde{S}^{z}_{i},\qquad
S^{z}_{N}=J+\widetilde{S}^{z}_{N{-}1},$ (S21)

for $i=2,...,N{-}1$. We can easily check that (S17) is fulfilled. In order to
invert this transformation and obtain the expression of the ghost variables
with respect to $S^{z}_{i}$, a convenient way is to write it in a matrix form

$\begin{bmatrix}S^{z}_{1}\\\ S^{z}_{2}\\\ S^{z}_{3}\\\ \vdots\\\
S^{z}_{N{-}2}\\\ S^{z}_{N{-}1}\\\ S^{z}_{N}\\\
\end{bmatrix}=\begin{bmatrix}-1&0&0&\ldots&0&0&1\\\ 1&-1&0&\ldots&0&0&1\\\
0&1&-1&\ldots&0&0&1\\\ \vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\vdots\\\
0&0&0&\ldots&-1&0&1\\\ 0&0&0&\ldots&1&-1&1\\\ 0&0&0&\ldots&0&1&1\\\
\end{bmatrix}\begin{bmatrix}\widetilde{S}^{z}_{1}\\\ \widetilde{S}^{z}_{2}\\\
\vdots\\\ \widetilde{S}^{z}_{N{-}3}\\\ \widetilde{S}^{z}_{N{-}2}\\\
\widetilde{S}^{z}_{N{-}1}\\\ J\\\ \end{bmatrix}.$ (S22)

The inverse transformation is given by

$\begin{bmatrix}\widetilde{S}^{z}_{1}\\\ \widetilde{S}^{z}_{2}\\\ \vdots\\\
\widetilde{S}^{z}_{N{-}3}\\\ \widetilde{S}^{z}_{N{-}2}\\\
\widetilde{S}^{z}_{N{-}1}\\\ J\\\
\end{bmatrix}=\dfrac{1}{N}\begin{bmatrix}-N{+}1&1&1&\ldots&1&1&1\\\
-N{+}2&-N{+}2&2&\ldots&2&2&2\\\ -N{+}3&-N{+}3&-N{+}3&\ldots&3&3&3\\\
\vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\vdots\\\
-2&-2&-2&\ldots&-2&N{-}2&N{-}2\\\ -1&-1&-1&\ldots&-1&-1&N{-}1\\\
1&1&1&\ldots&1&1&1\\\ \end{bmatrix}\begin{bmatrix}S^{z}_{1}\\\ S^{z}_{2}\\\
S^{z}_{3}\\\ \vdots\\\ S^{z}_{N{-}2}\\\ S^{z}_{N{-}1}\\\ S^{z}_{N}\\\
\end{bmatrix}.$ (S23)

Therefore, the ghost variables are given by the Jacobi momentum coordinates

$\widetilde{S}^{z}_{k}=\dfrac{k}{N}\sum_{i=1}^{N}S^{z}_{i}-\sum_{i=1}^{k}S^{z}_{i},\qquad\Delta\theta_{k}=\theta_{k+1}-\theta_{k},\qquad
k=1,...,N{-}1$ (S24)

as given in the main text. One can easily check that the transformation is
canonical with $\\{\theta,J\\}=1$,
$\\{\Delta\theta_{i},\widetilde{S}^{z}_{j}\\}=\delta_{ij}$ and
$\\{\theta,\widetilde{S}^{z}_{i}\\}=\\{J,\widetilde{S}^{z}_{i}\\}=\\{\theta,\Delta\theta_{i}\\}=\\{J,\Delta\theta_{i}\\}=\\{\widetilde{S}^{z}_{i},\widetilde{S}^{z}_{j}\\}=\\{\Delta\theta_{i},\Delta\theta_{j}\\}=0$
for all $i$ and $j$. In the same way, we find the change of coordinates for
the angles and relative angles.

In this new set of variables, the Hamiltonian
$H(J,\Delta\theta_{i},\widetilde{S}_{i}^{z})$ replacing the one in Eq. (S15)
is independent of the angle $\theta$.

### Beating variables and transformation back to Cartesian spin algebra

As mentioned in the main text, from canonical variables associated to a
conserved quantity $(\theta,J)$, one can construct beating observables
$A=f(J)\exp({-}{\rm i}\theta)$ and $A^{\dagger}=f(J)\exp({\rm i}\theta)$. One
can always perform the inverse change of coordinates from (S16) to (S14) by
going back to a spin algebra. For instance, using $f(J)=1$ we find the beating
variables

$A=\prod_{i=1}^{N}\dfrac{S^{-}_{i}}{\sqrt{S_{i}{}^{2}-(S_{i}^{z})^{2}}},\qquad
A^{*}=\prod_{i=1}^{N}\dfrac{S^{+}_{i}}{\sqrt{S_{i}{}^{2}-(S_{i}^{z})^{2}}}.$
(S25)

In the previous paragraph, we have described the canonical transformation from
$(\theta_{i},S_{i}^{z})$ to variables
$(\theta,J,\Delta\theta_{i},\widetilde{S}_{i}^{z})$. One can always go back to
a spin representation by performing the inverse change of coordinates from
(S16) to (S14). Here, this change of coordinates reads

$\displaystyle\widetilde{S}^{\pm}_{N}$
$\displaystyle=\sqrt{\widetilde{S}_{N}{}^{2}-J^{2}}\exp(\pm{\rm i}\theta),$
$\displaystyle\widetilde{S}^{z}_{N}$ $\displaystyle=J,$ (S26a)
$\displaystyle\widetilde{S}^{\pm}_{i}$
$\displaystyle=\sqrt{\widetilde{S}_{i}{}^{2}-(\widetilde{S}^{z}_{i})^{2}}\exp(\pm{\rm
i}\Delta\theta_{i}),$ $\displaystyle\widetilde{S}^{z}_{i}$
$\displaystyle=\widetilde{S}_{i}^{z},$ $\displaystyle i$
$\displaystyle=1,...,N{-}1,$ (S26b)

which allows us to obtain the spin algebra (S16). Note that
$A\propto\widetilde{S}_{N}^{-}$ and $A^{*}\propto\widetilde{S}_{N}^{+}$
corresponding to the beating variables associated with the conserved quantity
$J$ introduced in the main text.

To summarize, we started with a Hamiltonian in terms of a spin algebra in the
main text. Switching to canonical variables we obtained Hamiltonian (S15).
Then, using a canonical transformation, we found the new coordinates which
consist of the DOFs of the collective spin variables $(\theta,J)$ and the
ghost variables $(\Delta\theta_{i},\widetilde{S}_{i}^{z})$. From (S26), the
expression of the ghost variables with respect to the initial spin variables
read by means of the abbreviations
$S_{i}^{\rho}\equiv\sqrt{S_{i}^{2}-(S_{i}^{z})^{2}}$ and
$\widetilde{S}_{i}^{\rho}\equiv\sqrt{\widetilde{S}_{i}^{2}-(\widetilde{S}_{i}^{z})^{2}}$
(with Casimir invariants
$\widetilde{S}_{i}^{2}=|\widetilde{\mathbf{S}}_{i}|^{2}$)

$\displaystyle\widetilde{S}^{z}_{N}$
$\displaystyle=\dfrac{1}{N}\sum_{i=1}^{N}S_{i}^{z},$
$\displaystyle\widetilde{S}^{\pm}_{N}$
$\displaystyle=\frac{\widetilde{S}_{N}^{\rho}}{\prod_{i=1}^{N}S_{i}^{\rho}}\prod_{i=1}^{N}S_{i}^{\pm},$
(S27a) $\displaystyle\widetilde{S}^{z}_{k}$
$\displaystyle=\dfrac{k}{N}\sum_{i=1}^{N}S^{z}_{i}-\sum_{i=1}^{k}S_{i}^{z},$
$\displaystyle\widetilde{S}^{\pm}_{k}$
$\displaystyle=\dfrac{\widetilde{S}_{k}^{\rho}}{S_{k{+}1}^{\rho}\,S_{k}^{\rho}}S^{\pm}_{k{+}1}S^{\mp}_{k},$
$\displaystyle k$ $\displaystyle=1,...,N{-}1.$ (S27b)

This transformation is canonical, in the sense that it preserves the form of
the Poisson bracket (S13) for the spin algebra. The beating variables
associated with the conserved quantity $J$ are therefore
$A=\widetilde{S}_{N}^{-}$. We go from
$\\{S^{i}_{n},S^{j}_{m}\\}=\varepsilon_{ijk}S_{m}^{k}\delta_{nm}$, to
$\\{\widetilde{S}^{i}_{n},\widetilde{S}^{j}_{m}\\}=\varepsilon_{ijk}\widetilde{S}_{m}^{k}\delta_{nm}$.
Or equivalently, from $\\{S^{\pm}_{n},S^{z}_{m}\\}=\pm{\rm
i}S^{\pm}_{n}\delta_{nm}$ and $\\{S^{+}_{n},S^{-}_{m}\\}=-2{\rm
i}S^{z}_{n}\delta_{nm}$ to
$\\{\widetilde{S}^{\pm}_{n},\widetilde{S}^{z}_{m}\\}=\pm{\rm
i}\widetilde{S}^{\pm}_{n}\delta_{nm}$ and
$\\{\widetilde{S}^{+}_{n},\widetilde{S}^{-}_{m}\\}=-2{\rm
i}\widetilde{S}^{z}_{n}\delta_{nm}$. Therefore, the new spin variables
$\widetilde{\mathbf{S}}_{i}$ for $i=1,...,N{-}1$ are _ghost variables_ which
do not affect the DOF associated with the conserved quantity
$\widetilde{\mathbf{S}}_{N}$.

### Ghost and beating operators

From (S27a), we upgrade the beating and ghost variables to their quantum
mechanical counterparts, the beating and ghost operators. To do so, we replace
the variables by their associated operators (which are matrices) in the
expressions (S27a). We first notice that the quantities
$\mathbb{S}_{i}^{\rho}$, in the basis of the operators $\mathbb{S}_{i}^{z}$,
are diagonal matrices since $\mathbb{S}_{i}$ is proportional to identity
(Casimir invariant) and $\mathbb{S}_{i}^{z}$ is diagonal. This allows us to
compute the square roots and inverse of the operators $\mathbb{S}_{i}^{\rho}$.
In addition, these operators commute with each other. For the Heisenberg spin
model, we find a simplified version of a beating operator

$\mathbb{A}=\prod_{i{=}1}^{N}\mathbb{S}_{i}^{-},$ (S28)

since these operators are defined up to a factor. The expectation value of
this operator oscillates in time. From the classical ghost variables (S27b),
we obtain the ghost operators in the main text

$\displaystyle\widetilde{\mathbb{S}}^{z}_{k}$
$\displaystyle=\dfrac{k}{N}\sum_{i{=}1}^{N}\mathbb{S}^{z}_{i}-\sum_{i{=}1}^{k}\mathbb{S}_{j}^{z}$
$\displaystyle\widetilde{\mathbb{S}}^{\pm}_{k}$
$\displaystyle=\mathbb{D}_{k}\;\mathbb{S}^{\pm}_{k{+}1}\mathbb{S}^{\mp}_{k},\qquad\mathbb{D}_{k}=\dfrac{\widetilde{\mathbb{S}}_{k}^{\rho}}{\mathbb{S}_{k{+}1}^{\rho}\,\mathbb{S}_{k}^{\rho}}.$
(S29)

As mentioned above, all $\mathbb{S}_{k}^{\rho}$ are diagonal, which makes the
expression for $\mathbb{D}_{k}$ unique.

### Parameters for the Heisenberg model presented in the text

In Figs. 1 and 2, the self-energy $\omega_{i}$ of each spin site $i$ has been
generated randomly. The actual values and the matrices for nearest-neighbors
interactions, cf. Eq. (6) in the text, are

$[\omega_{i}]=\begin{bmatrix}0.6358\\\ 0.9452\\\ 0.2089\\\ 0.7093\\\ 0.2362\\\
0.1194\\\
\end{bmatrix},\quad[\sigma_{ij}]=\frac{1}{2}\begin{bmatrix}0&1&0&0&0&1\\\
1&0&1&0&0&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ 0&0&0&1&0&1\\\ 1&0&0&0&1&0\\\
\end{bmatrix},\quad[\Delta_{ij}]=\frac{1}{4}\begin{bmatrix}0&1&0&0&0&1\\\
1&0&1&0&0&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ 0&0&0&1&0&1\\\ 1&0&0&0&1&0\\\
\end{bmatrix}.$ (S30)

For the Lindblad operators in Fig. 2c, we have used a combination of ghost
operators such that
$\mathbb{L}_{\alpha_{ij}}=\gamma(\mathbb{S}_{i}^{z}-\mathbb{S}_{j}^{z})$. This
is done using (S21).

As initial condition we have taken $\rho(0)=\ket{\psi}\bra{\psi}$ with

$\ket{\psi}=\frac{\sum_{k_{1}\ldots k_{6}}\ket{k_{1}\ldots
k_{6}}C_{k_{1}\ldots k_{6}}}{\sqrt{\sum_{k_{1}\ldots k_{6}}C_{k_{1}\ldots
k_{6}}{}^{2}}}\quad\mbox{where}\quad C_{k_{1}\ldots k_{6}}=\\!\sum_{\xi=\pm
63}\\!\exp\big{(}{-}\big{[}\mbox{$\sum_{j}$}k_{j}\,2^{j}{-}\xi\big{]}^{2}/128\big{)}$
(S31)

and $k_{j}=\\{{-}\frac{1}{2},{+}\frac{1}{2}\\}$ and thus 64 different states
$\ket{k_{1}\ldots k_{6}}$.

##

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  * (4) H. Goldstein, Classical mechanics (Addison-Wesley, 1980).