Asher Roberts 11institutetext: Graduate Center, City University of New York,
New York, NY 10016, USA,
11email<EMAIL_ADDRESS>
WWW home page: https://asherbroberts.com

# The Rate of Convergence for Selberg’s Central Limit Theorem under the
Riemann Hypothesis

Asher Roberts

###### Abstract

We assume the Riemann hypothesis to improve upon the rate of convergence of
$(\log\log\log T)^{2}/\sqrt{\log\log T}$ in Selberg’s central limit theorem
for $\log|\zeta(1/2+it)|$ given by the author in [8]. We achieve a rate of
convergence of $\sqrt{\log\log\log\log T}/\sqrt{\log\log T}$ in the Dudley
distance. The proof is an adaptation of the techniques used by the author in
[8], based on the work of Radziwiłł and Soundararajan in [7] and Arguin et al.
in [1], combined with a lemma of Selberg [11] that provides for a mollifier
close to the critical line $\operatorname{Re}(s)=1/2$ under the Riemann
hypothesis.

###### keywords:

Probability · Analytic number theory · Riemann zeta function · Central limit
theorem

##  1 Introduction

Let $\tau$ be a random point uniformly distributed on $[T,2T]$. Selberg’s
central limit theorem states that as $T\to\infty$,

$\mathbb{P}\left(\log|\zeta(\tfrac{1}{2}+i\tau)|>V\sqrt{\tfrac{1}{2}\log\log
T}\right)\to\frac{1}{\sqrt{2\pi}}\int_{V}^{\infty}e^{-u^{2}/2}du,\quad\forall
V\in\mathbb{R}.$

In this paper, we determine the rate of convergence to a Gaussian random
variable in Selberg’s theorem using the Dudley (also called bounded
Wasserstein, e.g. see Gaunt and Li [2]) distance between two random vectors in
$\mathbb{R}^{n}$ defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ given by

$d_{\mathcal{D}}(X,Y)=\sup_{f\in\mathcal{L}}|\mathbb{E}[f(X)]-\mathbb{E}[f(Y)]|$

where $\mathcal{L}=\\{f:\mathbb{R}^{n}\to\mathbb{R}\mid f$ is Lipschitz,
$\|f\|_{\infty}\leq 1$, and $\|f\|_{\text{Lip}}\leq 1\\}$. Here and throughout
the paper $\|f\|_{\infty}$ denotes the supremum norm of $f$ and
$\|f\|_{\text{Lip}}=\sup_{x\neq y}\frac{|f(x)-f(y)|}{\|x-y\|_{1}}$ with
$\|\cdot\|_{1}$ denoting the $L^{1}$ norm. We use the Dudley distance since
the boundedness property allows us to use Lemma 2.12 in Proposition 1.6, while
the Lipschitz property assists us with the remaining propositions in the
paper.

###  1.1 Main Result

######  Theorem 1.1.

Assume the Riemann hypothesis and let $\tau$ be a random point distributed
uniformly on $[T,2T]$. Then for $T$ large enough,

$d_{\mathcal{D}}\left(\frac{\log|\zeta(\tfrac{1}{2}+i\tau)|}{\sqrt{\tfrac{1}{2}\log\log
T}},\mathcal{Z}\right)\ll\frac{\sqrt{\log\log\log\log T}}{\sqrt{\log\log T}},$

where $\mathcal{Z}$ is a standard normal random variable.

Theorem 1.1 is an improvement of the rate of convergence

$d_{\mathcal{D}}\left(\frac{\log|\zeta(\tfrac{1}{2}+i\tau)|}{\sqrt{\tfrac{1}{2}\log\log
T}},\mathcal{Z}\right)\ll\frac{(\log\log\log T)^{2}}{\sqrt{\log\log T}}$

given by the author in [8] using the Dudley distance. We modify the techniques
of the author in [8] to prove Theorem 1.1 by mollifying $\log|\zeta(1/2+it)|$
closer to the critical line $\operatorname{Re}(s)=1/2$. This allows us to
approximate $\log|\zeta(1/2+it)|$ by an “almost” Gaussian prime sum using
fewer steps, yielding a smaller error.

###  1.2 Relations to Previous Results

In 1946 Selberg presented a proof of his central limit theorem for the Riemann
zeta function in [9]. Later, Radziwiłł and Soundararajan used simpler
techniques to prove Selberg’s theorem for $\log|\zeta(\frac{1}{2}+it)|$ in
[7]. It is challenging to apply these techniques to the imaginary case, due to
the difficulty in mollifying the imaginary part of zeta. Nevertheless, the
mollifier used in Proposition 1.3 of this paper can be applied to the
imaginary part of zeta, but this would require a different way of moving the
imaginary part of zeta off axis than we present in the proof of Proposition
1.2. However, our work be extended to obtain analogous results for Dirichlet
$L$-functions, as Hsu and Wong did in [4] to prove the central limit theorem
using the techniques of Radziwiłł and Soundararajan.

Our paper mainly uses the techniques employed by the author in [8], although
some modifications are needed. Here the rate of convergence was obtained for
zeta at two points, while our result is presented for zeta at one point for
simplicity but can be similarly extended. The main difference lies in the
mollification step Proposition 1.3. Without the Riemann hypothesis, the
mollifier Selberg offers in [11] presents a sum over zeta zeros that is
difficult to estimate. By assuming the Riemann hypothesis and moving off axis
by $1/\log T$ (instead of by $(\log\log\log T)^{2}/\log T$ in [8]), we can
estimate this sum to be as close to order $1$ as we would like by further
iterating the log in the exponent of the parameter $X$ used.

In [10] Selberg claims a rate of convergence of $(\log\log\log
T)^{2}/\sqrt{\log\log T}$ unconditionally for the real part of zeta using the
Kolmogorov distance, a proof of which is given by Tsang in [12]. The author
reproduces this rate using the techniques of Radziwiłł and Soundararajan as
well as Arguin et al. in [8]. Selberg also claims a rate of $\log\log\log
T/\sqrt{\log\log T}$ for the imaginary case. An explicit proof of this result
is given by Wahl in [13] in the context of mod-gaussian convergence (more
information is given in Kowalski and Nikeghbali [6]).

While the Riemann hypothesis allows us to improve upon Selberg’s rate of
convergence in [10], there exist barriers towards further improvement.
Specifically, although we can push the parameter $X$ further in Proposition
1.3 as stated, we are still limited in the choice of parameters $X_{1}$ and
$X_{2}$ used in the Dirichlet polynomials we compare to Gaussian distributions
in Proposition 1.6. That is, iterating the log further in those parameters
results in Dirichlet polynomials that align less closely with Gaussian
distributions, worsening the errors presented in Lemmas 2.13 to 2.15.
Improving our overall stated rate of convergence therefore requires more
sophisticated techniques in handling parameters with log iterations that
approach order $1$.

###  1.3 Structure of the Proofs

The proof of Theorem 1.1 follows by five successive propositions. All proofs
are given in Section 2, and the distances between the random variables are
explicit at each step of the proof. The trickiest step is Proposition 1.3,
mollifying zeta. In particular, handling the sum over zeta zeros. However, the
step that incurs the greatest error is Proposition 1.5, truncating the prime
sum. Although we could improve the error obtained in mollifying by iterating
the logarithm further in the definition of the parameter $X$ used in
Proposition 1.3, we would still be required to perform the same truncation in
Proposition 1.5, in order to effectively compare moments with the Fourier
transforms in Proposition 1.6. That is, an improvement to the rate of
convergence we obtain requires an improvement in the way we truncate using
Proposition 1.5.

To connect zeta with the aforementioned “almost” Gaussian prime sum, we first
move off axis using Proposition 1.2, since the Riemann hypothesis will then
ensure that we do not encounter a zero when mollifying zeta in Proposition
1.3. To simplify notation, we set

$\sigma_{0}=\frac{1}{2}+\frac{1}{\log T},\qquad
s_{0}=\sigma_{0}+i\tau,\qquad\text{and}\qquad\mathfrak{s}=\sqrt{\tfrac{1}{2}\log\log
T}.$ (1)

The contribution towards the rate of convergence in Theorem 1.1 by moving off
axis is then given by

######  Proposition 1.2 (Moving off axis).

With the notation above, we have

$d_{\mathcal{D}}\left(\frac{\log|\zeta(\tfrac{1}{2}+i\tau)|}{\mathfrak{s}},\frac{\log|\zeta(s_{0})|}{\mathfrak{s}}\right)\ll\frac{1}{\sqrt{\log\log
T}}.$

Off axis, we are now able to approximate zeta with a mollifier in the form of
a Dirichlet polynomial $\mathcal{P}(s)$ given in Proposition 1.3. Note that
Proposition 1.3 is the only proposition conditional on the Riemann hypothesis.
The contribution towards the rate of convergence in Theorem 1.1 by mollifying
is then given by

######  Proposition 1.3 (Mollifying).

Assume the Riemann hypothesis. Let $X=T^{1/(\log_{4}T)^{1/4}}$ where
$\log_{\ell}$ stands for the logarithm iterated $\ell$ times, and

$\Lambda_{X}(n)=\begin{cases}\Lambda(n)&\text{for }1\leq n\leq X,\\\
\Lambda(n)\dfrac{\log\dfrac{X^{2}}{n}}{\log X}&\text{for }X\leq n\leq
X^{2}.\end{cases}$

Set

$\mathcal{P}(s_{0})=\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{n^{s_{0}}\log n}.$

Then

$d_{\mathcal{D}}\left(\frac{\log|\zeta(s_{0})|}{\mathfrak{s}},\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}}\right)\ll\frac{\sqrt{\log_{4}T}}{\sqrt{\log\log
T}}.$

We then discard the higher order primes in the sum $\mathcal{P}(s)$, whose
contribution towards the rate of convergence in Theorem 1.1 is given by

######  Proposition 1.4 (Discarding higher order primes).

Set

$\mathcal{P}(s_{0})=\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{n^{s_{0}}\log
n}\qquad\text{and}\qquad P(s_{0})=\operatorname{Re}\sum_{p\leq
X}\frac{1}{p^{s_{0}}}.$

With the notation above, we have

$d_{\mathcal{D}}\left(\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}},\frac{P(s_{0})}{\mathfrak{s}}\right)\ll\frac{1}{\sqrt{\log\log
T}}.$

In order to compare moments in Proposition 1.6, we truncate the prime sum. The
more we truncate the sum, the faster the sum tends toward a Gaussian random
variable. This truncation bears a contribution towards the rate of convergence
in Theorem 1.1 given by

######  Proposition 1.5 (Truncating the prime sum).

Set

$\displaystyle P_{1}(s_{0})$ $\displaystyle=\operatorname{Re}\sum_{p\leq
X_{1}}\frac{1}{p^{s_{0}}},$ $\displaystyle P_{2}(s_{0})$
$\displaystyle=\operatorname{Re}\sum_{X_{1}<p\leq
X_{2}}\frac{1}{p^{s_{0}}}\text{, and}$ $\displaystyle P_{3}(s_{0})$
$\displaystyle=\operatorname{Re}\sum_{X_{2}<p\leq X}\frac{1}{p^{s_{0}}},$

so that $P(s_{0})=P_{1}(s_{0})+P_{2}(s_{0})+P_{3}(s_{0})$ where
$X_{1}=T^{1/\log\log T}$ and $X_{2}=T^{1/\log\log\log T}$. With the notation
above, we have

$d_{\mathcal{D}}\left(\frac{P(s_{0})}{\mathfrak{s}},\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}}\right)\ll\frac{\sqrt{\log_{4}T}}{\sqrt{\log\log
T}}.$

Finally, we compare our truncated sum to a Gaussian random variable,
generating a contribution towards the rate of convergence in Theorem 1.1 given
by

######  Proposition 1.6.

With the notation above, we have

$d_{\mathcal{D}}\left(\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}},\mathcal{Z}\right)\ll\frac{1}{\sqrt{\log\log
T}},$

where $\mathcal{Z}$ is a standard normal random variable.

Acknowledgements. I give thanks to Emma Bailey for spotting a crucial issue in
one of my initial drafts for this paper, and to Prof. Louis-Pierre Arguin for
his helpful insight that provided me with the confidence to fix it. I would
also like to thank everyone who read through the paper and offered edits. Your
feedback is very much appreciated.
Partial support is provided by grants NSF CAREER 1653602 and NSF DMS 2153803.

##  2 Proofs

###  2.1 Proof of Theorem 1.1

######  Proof 2.1.

With the notation above, we have by Propositions 1.2 to 1.6 and the triangle
inequality,

$\displaystyle
d_{\mathcal{D}}\left(\frac{\log|\zeta(\tfrac{1}{2}+i\tau)|}{\sqrt{\tfrac{1}{2}\log\log
T}},\mathcal{Z}\right)$ $\displaystyle\ll
d_{\mathcal{D}}\left(\frac{\log|\zeta(\tfrac{1}{2}+i\tau)|}{\mathfrak{s}},\frac{\log|\zeta(s_{0})|}{\mathfrak{s}}\right)$
$\displaystyle\phantom{\ll}+d_{\mathcal{D}}\left(\frac{\log|\zeta(s_{0})|}{\mathfrak{s}},\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}}\right)$
$\displaystyle\phantom{\ll}+d_{\mathcal{D}}\left(\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}},\frac{P(s_{0})}{\mathfrak{s}}\right)$
$\displaystyle\phantom{\ll}+d_{\mathcal{D}}\left(\frac{P(s_{0})}{\mathfrak{s}},\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}}\right)$
$\displaystyle\phantom{\ll}+d_{\mathcal{D}}\left(\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}},\mathcal{Z}\right)$
$\displaystyle\ll\frac{\sqrt{\log\log\log\log T}}{\sqrt{\log\log T}}.$

###  2.2 Proof of Proposition 1.2

######  Proof 2.2.

Denote $\log|\zeta(\tfrac{1}{2}+i\tau)|/\mathfrak{s}$ by $\mathcal{V}$ and
$\log|\zeta(s_{0})|/\mathfrak{s}$ by $\mathcal{W}$. Then

$d_{\mathcal{D}}(\mathcal{V},\mathcal{W})=\sup_{f\in\mathcal{L}}|\mathbb{E}[f(\mathcal{V})]-\mathbb{E}[f(\mathcal{W})]|\leq\sup_{f\in\mathcal{L}}\mathbb{E}[|f(\mathcal{V})-f(\mathcal{W})|].$

Since $f:\mathbb{R}\to\mathbb{R}$ where $f\in\mathcal{L}$ is Lipschitz with
$\|f\|_{\text{Lip}}\leq 1$, we have
$|f(v)-f(w)|\leq\|f\|_{\text{Lip}}\|v-w\|_{1}\leq\|v-w\|_{1}$, and thus

$|f(\mathcal{V})-f(\mathcal{W})|\leq\|\mathcal{V}-\mathcal{W}\|_{1}.$ (2)

Therefore, by definition of the Dudley distance, the above distance is less
than or equal to $\mathbb{E}[|\mathcal{V}-\mathcal{W}|]$. We now introduce
Proposition 1 of [7] as Lemma 2.3.

######  Lemma 2.3.

With the notation above, for $T$ large enough and any $\sigma>1/2$, we have

$\mathbb{E}\Big{[}\Big{|}\log|\zeta(\tfrac{1}{2}+i\tau)|-\log|\zeta(\sigma+i\tau)|\Big{|}\Big{]}\ll(\sigma-\tfrac{1}{2})\log
T.$

The proof of Lemma 2.3 uses Hadamard’s factorization formula for the completed
$\zeta$-function and is given in [7]. By Lemma 2.3,
$\mathbb{E}[|\mathcal{V}-\mathcal{W}|]\ll 1/\sqrt{\log\log T}$, and so our
result follows.

###  2.3 Proof of Proposition 1.3

######  Proof 2.4.

Similar to the way we proved Proposition 1.2, we denote
$\log|\zeta(s_{0})|/\mathfrak{s}$ by $\mathcal{W}$,
$\operatorname{Re}\mathcal{P}(s_{0})/\mathfrak{s}$ by $\mathcal{X}$ and have

$d_{\mathcal{D}}(\mathcal{W},\mathcal{X})=\sup_{f\in\mathcal{L}}|\mathbb{E}[f(\mathcal{W})]-\mathbb{E}[f(\mathcal{X})]|\leq\sup_{f\in\mathcal{L}}\mathbb{E}[|f(\mathcal{W})-f(\mathcal{X})|].$
(3)

To compute this distance, we first approximate zeta using a mollifier of
Selberg [11] given by the following lemma.

######  Lemma 2.5.

With the same assumptions as Proposition 1.3, and for $s\neq 1$, $s\neq\rho$,
and $s\neq-2q$ $(q=1,2,3,\ldots)$, where $\rho$ is a zero of $\zeta$, we have

$\frac{\zeta^{\prime}}{\zeta}(s)=-\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{n^{s}}+\frac{1}{\log
X}\sum_{\rho}\frac{X^{\rho-s}-X^{2(\rho-s)}}{(s-\rho)^{2}}+O(1).$

######  Proof 2.6.

The proof of the result

$\displaystyle\frac{\zeta^{\prime}}{\zeta}(s)=-\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{n^{s}}$
$\displaystyle+\frac{X^{2(1-s)}-X^{1-s}}{\log X\cdot(1-s)^{2}}+\frac{1}{\log
X}\sum_{q=1}^{\infty}\frac{X^{-2q-s}-X^{-2(2q+s)}}{(2q+s)^{2}}$
$\displaystyle+\frac{1}{\log
X}\sum_{\rho}\frac{X^{\rho-s}-X^{2(\rho-s)}}{(s-\rho)^{2}}$

is given unconditionally in Lemma 2 of Selberg [11] using a contour integral.
For $s=\sigma+it$ we have $|1-s|^{2}\asymp t^{2}$, and so

$\frac{X^{2(1-s)}-X^{1-s}}{\log
X\cdot(1-s)^{2}}\ll\frac{X^{2(1-\sigma)}+X^{1-\sigma}}{t^{2}\log
X}\ll\frac{X}{t^{2}\log X}\ll\frac{1}{T\log T},$

while the sum over $q$ is clearly $O(1)$.

We continue by applying the Riemann hypothesis, setting $s=\sigma+it$ and
$\rho=1/2+i\gamma$, while integrating both sides of Lemma 2.5 to obtain

$\int_{\sigma_{0}}^{2}\operatorname{Re}\frac{\zeta^{\prime}}{\zeta}(s)\,d\sigma=-\operatorname{Re}\sum_{n<X^{2}}\Lambda_{X}(n)\int_{\sigma_{0}}^{2}n^{-s}d\sigma+\frac{1}{\log
X}\operatorname{Re}\sum_{\rho}\int_{\sigma_{0}}^{2}\frac{X^{\rho-s}-X^{2(\rho-s)}}{(s-\rho)^{2}}\,d\sigma.$

The first summand becomes

$\displaystyle-\operatorname{Re}\sum_{n<X^{2}}\Lambda_{X}(n)\int_{\sigma_{0}}^{2}n^{-\sigma-
it}d\sigma$
$\displaystyle=-\operatorname{Re}\sum_{n<X^{2}}\Lambda_{X}(n)n^{-it}\left[\frac{\exp(-\sigma\log
n)}{-\log n}\right]_{\sigma=\sigma_{0}}^{\sigma=2}$
$\displaystyle=-\operatorname{Re}\sum_{n<X^{2}}\Lambda_{X}(n)n^{-it}\left[\frac{n^{-\sigma}}{-\log
n}\right]_{\sigma=\sigma_{0}}^{\sigma=2}$
$\displaystyle=-\operatorname{Re}\sum_{n<X^{2}}\Lambda_{X}(n)\left[\frac{n^{-s_{0}}}{\log
n}-\frac{n^{-2-it}}{\log n}\right]$
$\displaystyle=-\operatorname{Re}\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{\log
n}n^{-s_{0}},$

by the summability of $\sum n^{-2-it}$. Meanwhile, the second summand is

$\displaystyle\frac{1}{\log
X}\operatorname{Re}\sum_{\rho}\int_{\sigma_{0}}^{2}\frac{X^{\rho-s}-X^{2(\rho-s)}}{(s-\rho)^{2}}\,d\sigma$
$\displaystyle\ll\frac{1}{\log
X}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}\int_{\sigma_{0}}^{2}(X^{1/2-\sigma}+X^{2(1/2-\sigma)})\,d\sigma$
$\displaystyle\ll\frac{1}{(\log X)^{2}}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}.$

Therefore,

$\log|\zeta(2+it)|-\log|\zeta(s_{0})|=-\operatorname{Re}\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{\log
n}n^{-s_{0}}+O\left(\frac{1}{(\log
X)^{2}}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}\right).$

As we obtained the inequality (2) by taking advantage of the Lipschitz nature
of $f$, we similarly obtain

$|f(\mathcal{W})-f(\mathcal{X})|\leq\|\mathcal{W}-\mathcal{X}\|_{1},$

and so the expectation in (3) is less than or equal to

$\displaystyle\mathbb{E}[|\mathcal{W}-\mathcal{X}|]$
$\displaystyle=\frac{1}{\mathfrak{s}}\mathbb{E}[|\log|\zeta(s_{0})|-\operatorname{Re}\mathcal{P}(s_{0})|]$
$\displaystyle\ll\frac{1}{\mathfrak{s}}\mathbb{E}\left[\frac{1}{(\log
X)^{2}}\sum_{\rho}\frac{1}{|s_{0}-\rho|^{2}}\right],$

since $\log|\zeta(2+it)|=O(1)$. We then proceed similarly as in the proof of
Proposition 1 of Harper [3], computing the expectation by integrating

$\displaystyle\int_{T}^{2T}\sum_{\rho:T\leq\gamma\leq
2T}\frac{1}{|s_{0}-\rho|^{2}}\,dt$ $\displaystyle=\sum_{\rho:T\leq\gamma\leq
2T}\int_{T}^{2T}\frac{1}{|s_{0}-\rho|^{2}}\,dt.$

Since we are assuming the Riemann hypothesis, and since we have moved off axis
by $1/\log T$, we have $|s_{0}-\rho|>1/\log T$ for all $\rho$, and therefore
the integral above is

$\displaystyle\leq\sum_{\rho:T\leq\gamma\leq 2T}\int_{1/\log
T}^{\infty}\frac{1}{t^{2}}\,dt=\log T\sum_{\rho:T\leq\gamma\leq 2T}1\ll(\log
T)^{2},$

where in the end we used the Riemann-von Mangoldt formula (see, e.g., Ivić
[5]), which provides an estimate of $\log T$ for the number of zeta zeros
between $T$ and $2T$. Thus

$\frac{1}{\mathfrak{s}}\mathbb{E}[|\log|\zeta(s_{0})|-\operatorname{Re}\mathcal{P}(s_{0})|]\ll\frac{1}{\mathfrak{s}}\cdot\frac{1}{(\log
X)^{2}}\cdot(\log T)^{2}\ll\frac{\sqrt{\log_{4}T}}{\sqrt{\log\log T}}$

as desired.

###  2.4 Proof of Proposition 1.4

######  Proof 2.7.

By taking advantage of the Lipschitz nature of the Dudley distance, we obtain

$\displaystyle
d_{\mathcal{D}}\left(\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}},\frac{P(s_{0})}{\mathfrak{s}}\right)$
$\displaystyle\leq\mathbb{E}\left[\left|\frac{\operatorname{Re}\mathcal{P}(s_{0})}{\mathfrak{s}}-\frac{P(s_{0})}{\mathfrak{s}}\right|\right].$

Then $\Lambda_{X}(n)\leq\Lambda(n)$, and so

$\sum_{n<X^{2}}\frac{\Lambda_{X}(n)}{n^{s_{0}}\log
n}\ll\sum_{n<X}\frac{\Lambda(n)}{n^{s_{0}}\log n},$

since $T^{2/(\log_{4}T)^{1/4}}=O(T^{1/(\log_{4}T)^{1/4}})$. Therefore,
computing the difference in the expectation we get

$\frac{1}{\mathfrak{s}}|\operatorname{Re}\mathcal{P}(s_{0})-P(s_{0})|\ll\frac{1}{\mathfrak{s}}\left|\sum_{2\leq
p^{2}\leq
X}\frac{1}{2p^{2s_{0}}}\right|+\frac{1}{\mathfrak{s}}\left|\sum_{\begin{subarray}{c}2\leq
p^{k}\leq X,\\\ k\geq 3\end{subarray}}\frac{\log p}{p^{ks_{0}}(k\log
p)}\right|.$ (4)

Then since

$\left|\sum_{\begin{subarray}{c}2\leq p^{k}\leq X,\\\ k\geq
3\end{subarray}}\frac{\log p}{p^{ks_{0}}(k\log
p)}\right|\leq\sum_{\begin{subarray}{c}2\leq p^{k}\leq X,\\\ k\geq
3\end{subarray}}\frac{1}{3p^{k\sigma_{0}}}=O(1)$

and

$\displaystyle\mathbb{E}\left[\left|\sum_{2\leq p^{2}\leq
X}\frac{1}{2p^{2(s_{0})}}\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\left|\sum_{2\leq p^{2}\leq
X}\frac{1}{2p^{2(s_{0})}}\right|^{2}\right]^{1/2}$
$\displaystyle=\frac{1}{T}\int_{T}^{2T}\left|\sum_{2\leq p^{2}\leq
X}\frac{1}{2p^{2(s_{0})}}\right|^{2}dt$
$\displaystyle=\frac{1}{T}\cdot\frac{1}{4}\sum_{p_{1},p_{2}\leq\sqrt{X}}\int_{T}^{2T}\frac{1}{p_{1}^{2(\sigma_{0}+it)}p_{2}^{2(\sigma_{0}-it)}}\,dt$
$\displaystyle\ll\sum_{p\leq\sqrt{X}}\frac{1}{p^{4\sigma_{0}}}+\frac{1}{T}\sum_{\begin{subarray}{c}p_{1},p_{2}\leq\sqrt{X}\\\
p_{1}\neq
p_{2}\end{subarray}}\frac{1}{p_{1}^{2\sigma_{0}}p_{2}^{2\sigma_{0}}}\sqrt{p_{1}p_{2}}=O(1),$

we see that (4) is $\ll 1/\mathfrak{s}$.

###  2.5 Proof of Proposition 1.5

######  Proof 2.8.

We have

$\displaystyle
d_{\mathcal{D}}\left(\frac{P(s_{0})}{\mathfrak{s}},\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}}\right)\leq\mathbb{E}\left[\left(\frac{P_{3}(s_{0})}{\mathfrak{s}}\right)^{2}\right]^{1/2}.$

To evaluate this expectation, we use Lemma 5 of [8], which we introduce as
Lemma 2.9

######  Lemma 2.9.

For any non-negative integer $k$ we have

$\displaystyle\mathbb{E}\left[\left(\operatorname{Re}\left(\sum_{p\leq
X}\frac{1}{p^{s_{0}}}\right)\right)^{2k}\right]$
$\displaystyle=2^{-k}\frac{(2k)!}{k!}\left[\left(\frac{1}{2}\sum_{p\leq
X}\frac{1}{p^{2\sigma_{0}}}\right)^{k}+O_{k}\left(\frac{1}{2}\sum_{p\leq
X}\frac{1}{p^{2\sigma_{0}}}\right)^{k-2}\right]+O\left(\frac{X^{4k}}{T}\right),$

where $O_{k}$ indicates that the implied constant depends on $k$. If $k$ is
odd, we have

$\mathbb{E}\left[\left(\operatorname{Re}\left(\sum_{p\leq
X}\frac{1}{p^{s_{0}}}\right)\right)^{k}\right]=O\left(\frac{X^{2k}}{T}\right).$

######  Proof 2.10.

The statement follows from the proof of Lemma 5 of [8] by taking
$u=1,u^{\prime}=0$.

By Lemma 2.9, we have

$\mathbb{E}[|P_{3}(\tfrac{1}{2}+it)|^{2k}]\ll\frac{(2k)!}{k!2^{k}}\left(\sqrt{\tfrac{1}{4}\log_{4}T}\right)^{2k}+O\left(\frac{X^{4k}}{T}\right).$
(5)

Here we used Mertens’s theorem to estimate

$\sum_{X_{2}<p\leq X}\frac{1}{p}\ll\log\log X_{2}-\log\log
X=\log(\sqrt{\log\log\log
T})-\log((\log_{4}T)^{1/4})\ll\tfrac{1}{2}\log_{4}T.$

Thus the expectation above is less than or equal to
$\frac{1}{\mathfrak{s}}(\sqrt{\log_{4}T}+O(1))$ when $k=1$.

###  2.6 Proof of Proposition 1.6

######  Proof 2.11.

Similar to Proposotion 6 of the author in [8], we obtain the error in
approximating $P_{1}(s_{0})+P_{2}(s_{0})$ with a Gaussian random variable by
comparing their Fourier transforms using Lemma 3.11 of [1], which we introduce
as Lemma 2.12,

######  Lemma 2.12.

Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}$ with Fourier
transform $\hat{\mu}$ and $\hat{\nu}$. There exists constant $c>0$ such that
for any function $f:\mathbb{R}\to\mathbb{R}$ with Lipschitz constant $1$ and
for any $R,F>0$,

$\left|\int_{\mathbb{R}}f\,d\mu-\int_{\mathbb{R}}f\,d\nu\right|\leq\frac{1}{F}+\|f\|_{\infty}\left\\{(RF)\|(\hat{\mu}-\hat{\nu})\mathbbm{1}_{(-F,F)}\|_{\infty}+\mu((-R,R)^{c})+\nu((-R,R)^{c})\right\\}.$

To use Lemma 2.12, we introduce a lemma that shows that the difference in
expectations between the Fourier transforms of the prime sum and that of a
Gaussian random variable is small.

######  Lemma 2.13.

$\displaystyle\left|\mathbb{E}\left[\exp\left(\frac{i\xi
P_{1}(s_{0})}{\mathfrak{s}}\right)\right]-\mathbb{E}[\exp(i\xi\mathcal{Z})]\right|$
$\displaystyle\ll\frac{1}{(\log\log T)^{2}}.$

######  Proof 2.14.

We first truncate the Fourier transform of $P_{1}(s_{0})$ for
$\xi\in\mathbb{R}$ by writing

$\displaystyle\mathbb{E}\left[\exp\left(\frac{i\xi
P_{1}(s_{0})}{\mathfrak{s}}\right)\cdot\mathbbm{1}(|P_{1}(s_{0})|\leq\log\log
T)\right]$ $\displaystyle\ll\sum_{n\leq
N}\frac{1}{n!}i^{n}\mathbb{E}\left[\left(\frac{\xi
P_{1}(s_{0})}{\mathfrak{s}}\right)^{n}\cdot\mathbbm{1}(|P_{1}(s_{0})|\leq\log\log
T)\right]+(\log T)^{-C}$

for a constant $C$, where the error term $(\log T)^{-C}$ is estimated in [8]
using Stirling’s formula. The event $\\{|P_{1}(s_{0})|\leq\log\log T\\}$ is
introduced to assist with the truncation, and is subsequently removed by
inserting the complement of this event, $\\{|P_{1}(s_{0})|>\log\log T\\}$, and
showing that the probability of the complement is small. After truncating, the
moment estimate for $P_{1}$ in Lemma 2.9 is used to provide the error in
comparing the truncated moments to a Gaussian. Full details of the proof are
given in Lemma 9 of [8].

Next we introduce Lemma 2.15 as an analog of Lemma 2.13 for $P_{2}(s_{0})$.

######  Lemma 2.15.

$\displaystyle\left|\mathbb{E}\left[\exp\left(\frac{i\xi
P_{2}(s_{0})}{\mathfrak{s}}\right)\right]-\mathbb{E}[\exp(i\xi\mathcal{Z})]\right|$
$\displaystyle\ll\frac{1}{(\log\log T)^{2}}.$

######  Proof 2.16.

We truncate the Fourier transform of $P_{2}(s_{0})$ for $\xi\in\mathbb{R}$ by
writing

$\displaystyle\mathbb{E}\left[\exp\left(\frac{i\xi
P_{2}(s_{0})}{\mathfrak{s}}\right)\cdot\mathbbm{1}(|P_{2}(s_{0})|\leq
2\log\log\log T)\right]$ $\displaystyle\ll\sum_{n\leq
N}\frac{1}{n!}i^{n}\mathbb{E}\left[\left(\frac{\xi
P_{2}(s_{0})}{\mathfrak{s}}\right)^{n}\cdot\mathbbm{1}(|P_{2}(s_{0})|\leq
2\log\log\log T)\right]+(\log T)^{-D}$

for a constant $D$, similar to the way we truncated the Fourier transform of
$P_{1}(s_{0})$. Note that we take $2\log\log\log T$ in the event in order to
achieve an error of $1/(\log\log T)^{2}$, since the proof requires an estimate
for $\mathbb{P}(|P_{2}(s_{0})|>2\log\log\log T)$ instead of for
$\mathbb{P}(|P_{1}(s_{0})|>\log\log T)$. As such,

$\displaystyle\mathbb{P}(|P_{2}(s_{0})|>2\log\log\log T)$
$\displaystyle\leq\frac{1}{(2\log\log\log
T)^{2k}}\mathbb{E}\left[|P_{2}(s_{0})|^{2k}\right]$
$\displaystyle\ll\frac{1}{(2\log\log\log
T)^{2k}}\left[\frac{(2k)!}{k!2^{k}}\left(\sqrt{\tfrac{1}{2}\log\log\log
T}\right)^{2k}\right]$ $\displaystyle\ll\frac{1}{(2\log\log\log
T)^{2k}}\left[\frac{(2k)^{2k}}{k^{k}2^{k}}\frac{\exp(-2k)}{\exp(-k)}\left(\sqrt{\tfrac{1}{2}\log\log\log
T}\right)^{2k}\right]$ $\displaystyle=\frac{1}{(2\log\log\log
T)^{2k}}\left[(2k)^{k}\exp(-k)2^{-k}(\log\log\log T)^{k}\right],$

so we can set $k=\lfloor\log\log\log T\rfloor$ to get

$\displaystyle\mathbb{P}(|P_{2}(s_{0})|>2\log\log\log T)$
$\displaystyle\ll\frac{1}{(2k)^{2k}}[(2k)^{k}\exp(-k)2^{-k}k^{k}]$
$\displaystyle=2^{-2k}\exp(-k)$ $\displaystyle\ll\exp(-2k)=\frac{1}{(\log\log
T)^{2}}.$

The rest of the proof is the same as for Lemma 2.13.

Now, using Lemmas 2.12 to 2.15 we have

$\displaystyle
d_{\mathcal{D}}\left[\left(\frac{P_{1}(s_{0})+P_{2}(s_{0})}{\mathfrak{s}}\right),\mathcal{Z}\right]\leq
d_{\mathcal{D}}\left[\left(\frac{P_{1}(s_{0})}{\mathfrak{s}}\right),\mathcal{Z}\right]+d_{\mathcal{D}}\left[\left(\frac{P_{2}(s_{0})}{\mathfrak{s}}\right),\mathcal{Z}\right]$
$\displaystyle\leq\frac{1}{F}+\left\\{(R_{1}F)\left|\mathbb{E}\left[\exp\left(\frac{i\xi
P_{1}(s_{0})}{\mathfrak{s}}\right)\right]-\mathbb{E}[\exp(i\xi\mathcal{Z})]\right|\right.$
$\displaystyle\phantom{\leq}\left.+\mathbb{P}\left(\left|\frac{P_{1}(s_{0})}{\mathfrak{s}}\right|>R_{1}\right)+\mathbb{P}(|\mathcal{Z}|>R_{1})\right\\}$
$\displaystyle\phantom{\leq}+\frac{1}{F}+\left\\{(R_{2}F)\left|\mathbb{E}\left[\exp\left(\frac{i\xi
P_{2}(s_{0})}{\mathfrak{s}}\right)\right]-\mathbb{E}[\exp(i\xi\mathcal{Z})]\right|\right.$
$\displaystyle\phantom{\leq}\left.+\mathbb{P}\left(\left|\frac{P_{2}(s_{0})}{\mathfrak{s}}\right|>R_{2}\right)+\mathbb{P}(|\mathcal{Z}|>R_{2})\right\\}$

for any $R_{1},R_{2},F>0$. Since

$\displaystyle\mathbb{P}\left(|P_{2}(s_{0})|>\sqrt{\log\log\log
T}\sqrt{\log\log T}\right)$ $\displaystyle\leq\frac{1}{(\log\log\log
T)(\log\log T)}\mathbb{E}[|P_{2}(s_{0})|^{2}]=\frac{1}{\log\log T}$

we can take $R_{1}=\sqrt{\log\log T}$, $R_{2}=\sqrt{\log\log\log T}$ and
$F=C\sqrt{\log\log T}$ to get

$\displaystyle\frac{1}{\sqrt{\log\log T}}+\left[C\log\log
T\left(\frac{1}{(\log\log T)^{2}}\right)+\frac{1}{\log T}+\frac{1}{(\log
T)^{D}}\right]$ $\displaystyle+\frac{1}{\sqrt{\log\log
T}}+\left[C\sqrt{\log\log\log T}\sqrt{\log\log T}\left(\frac{1}{(\log\log
T)^{2}}\right)+\frac{1}{\log\log T}+\frac{1}{(\log\log T)^{D}}\right]$
$\displaystyle\ll\frac{1}{\sqrt{\log\log T}}.$

## References

  * [1] Arguin, L.-P., Belius, D., Bourgade, P., Radziwiłł, M., Soundararajan, K.: Maximum of the Riemann zeta function on a short interval of the critical line. Preprint, arxiv: 1612.08575v2 (2017)
  * [2] Gaunt, R.E., Li, S.: Bounding Kolmogorov distances through Wasserstein and related integral probability metrics. (2022). Preprint, arXiv:2201.12087.
  * [3] Harper, A.J.: A note on the maximum of the Riemann zeta function, and log-correlated random variables. Preprint, arxiv: 1304.0677 (2013)
  * [4] Hsu, P., Wong, P.: On Selberg’s central limit theorem for Dirichlet L-functions. Journal de Théorie des Nombres de Bordeaux. 32, 685-710 (2020)
  * [5] Ivić, A.: The Riemann Zeta Function. Wiley. (1985)
  * [6] Kowalski, E., Nikeghbali, A.: Mod-Gaussian convergence and the value distribution of $\zeta(1/2+it)$ and related quantities. J. London Math. Soc. 86(1), 291-319 (2012)
  * [7] Radziwiłł, M., Soundararajan, K.: Selberg’s central limit theorem for $\log|\zeta(\frac{1}{2}+it)|$. Enseign. Math. 63(1-2), 1-19 (2017)
  * [8] Roberts, A.: The multivariate rate of convergence for Selberg’s central limit theorem. Preprint, arxiv: 2212.01411 (2022)
  * [9] Selberg, A.: Contributions to the theory of the Riemann zeta-function. Arch. Math. Naturvid. 48(5), 89-155 (1946)
  * [10] Selberg, A.: Old and new conjectures and results about a class of Dirichlet series. Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989). Univ. Salerno, Salerno. 367-385 (1992)
  * [11] Selberg, A.: On the normal density of primes in small intervals, and the difference between consecutive primes. Arch. Math. Naturvid. 47(6), 87-105 (1943)
  * [12] Tsang, K.: The distribution of the values of the Riemann zeta-function. Thesis Princeton University (1984)
  * [13] Wahl, M.: On the mod-Gaussian convergence of a sum over primes. Mathematische Zeitschrift. 276(3), 635-654 (2014)


# Encoding Binary Events from Continuous Time Series in Rooted Trees using
Contrastive Learning

Tobias Engelhardt Rasmussen Corresponding Author<EMAIL_ADDRESS>Technical
University of Denmark  Siv Sørensen  Technical University of Denmark

###### Abstract

Broadband infrastructure owners do not always know how their customers are
connected in the local networks, which are structured as rooted trees. A
recent study is able to infer the topology of a local network using discrete
time series data from the leaves of the tree (customers). In this study we
propose a contrastive approach for learning a binary event encoder from
continuous time series data. As a preliminary result, we show that our
approach has some potential in learning a valuable encoder.

##  1 Introduction

Hybrid-Fiber Coaxial (HFC) networks are used to connect users to the internet
using coaxial cables in which data is transmitted using radio frequency (RF)
signals. Each HFC network is connected to the optical backbone grid through a
local singular conversion node, called a CMC, where the signal is converted
from coaxial to optical and vice versa. Within each HFC network each customer
is connected to the CMC through a sequence of cable amplifiers and cable
splitters arranged in a tree-like structure. Here, the customers represent the
leaf nodes, the cable splitters the internal nodes, and the CMC the root node
[8].

Due to digitization, complexity, and time, some degree of the cabling,
hereinafter referred to as the topology, of most HFC networks is unknown to
the infrastructure owner. This is problematic since knowing the topology is
crucial for network maintenance staff when localizing and resolving network
faults. Incomplete topology records on average lead to a significant increase
in both resolution and driving time for the maintenance staff.

A recent study, currently under review, shows that it is possible to infer a
missing topology with high accuracy using discrete time series data collected
only at leaf level [9]. The authors do this by considering both a Maximum
Parsimony optimality criterion [2] as well as geographic distances between
components of the network. The most parsimonious tree topology is the topology
which can explain the observed leaf data in terms of fewest data mutations.
I.e. given the RF signal is initially transmitted through a singular root
node, one can reconstruct the most probable evolution of the signal throughout
the network branches at each time step and count the number of changes [2, 4].
This number is often referred to as the parsimony score of a proposed tree
topology. However, this approach relies both on the observed data being
discrete, as well as distinguishable data mutations, hereinafter events,
occurring on most branches of the network over time.

Our work focuses on extracting significant discrete events from continuous
time series data, a task which, in this setting, is unsolved. For simplicity
we initially consider binary events. Due to the parsimony score counting the
number of mutations needed to explain the observed leaf data, learning an
encoder using the parsimony score as a loss function is not feasible. No
mutations are needed to explain identical data, so the model will simply
encode the same data point for all leaf nodes.

In this work we propose using the parsimony score to train a binary event
encoder using a contrastive approach [6]. As a preliminary result, we aim to
analyse the feasibility of the proposed approach using simulated (perfect)
data, where all possible topologies are known and unique events are simulated
on all network edges. We will let this approach be the foundation for future
work.

##  2 Proposed Method

We propose a contrastive approach building on a Siamese network[5] where we
will let the discrimination be determined by the parsimony score. Our approach
consists of a Siamese network that encodes the continuous time series of all
the modems into a sequences of bits. The parsimony score is calculated across
multiple topologies and used to minimize the score of the true topology while
simultaneously maximising the difference in parsimony scores using different
topologies. Our approach is illustrated in Appendix A. For the encoder model
we propose using a causal convolutional deep neural network inspired by
Franceschi et. al in [3]. However, we do not pool the time dimension (the
desired output is a time series), but instead send each time point through a
sigmoid-function.

Let $\mathcal{P}(\mathbf{X},G)$ be the vector of parsimony scores for each
time point/step of the encoded bits $\mathbf{X}$ using the graph $G$. We
initially propose the loss function:

$\begin{split}&L(\mathbf{X},G^{p},G_{1}^{n},\dots
G_{K}^{n})=\sum\mathcal{P}(\mathbf{X},G^{p})\ +\\\
&\frac{\alpha}{2}\sqrt{\sum_{g_{1}\in\mathcal{G}}\sum_{g_{2}\in\mathcal{G}\backslash
g_{1}}\frac{1}{\left(\|\mathcal{P}(\mathbf{X},g_{1})-\mathcal{P}(\mathbf{X},g_{2})\|_{2}^{2}+1\right)^{2}}}\end{split}$
(1)

where $G^{p}$ is the true topology, $G^{n}_{\bullet}$ are the $K$ negative
(sampled) topologies, and $\mathcal{G}=\\{G^{p},G^{n}_{1},\dots,G^{n}_{K}\\}$.
This loss function considers the difference between resulting parsimony scores
using all pairwise topology samples, rewarding large score differences and
punishing small ones.

##  3 Evaluation

In order to validate the feasibility of the proposed algorithm we run it on
simulated data and present the results.

For a tree with four internal nodes whereof one is a root node, there exist 16
different permutations of the topology. All possible permutations of the tree
are provided in the Appendix B. We simulate both the network topology and
number of modems for each splitter along with the continuous time series leaf
data. We use the AR(2) model[7] as a base signal inspired by [1]. We create
perfectly distinguishable events by splitting the time points into chunks
(with a start and end time) and for each chunk we sample an internal edge and
modify the signal of the affected modems, below this edge in the network,
accordingly. The sampling procedure is described in detail in the Appendix C.

We simulate 1,000 observations whereof 500 will be used as a separate test
set. For each observation we simulate time series of length 4,000 that we
split into 40 chunks for simulating events. We train the model experimentally
setting the weight-parameter to $\alpha=2.0$. 1(a) shows the fraction of
observations for which the true topology leads to the lowest parsimony score,
respectively, the second and third lowest parsimony score of all possible
topologies.

Table 1: Results of our encoded model

(a) Fraction of observations in the test set for which the true topology leads to the lowest parsimony score. Top-1 | Top-2 | Top-3
---|---|---
0.748 | 0.956 | 0.998

(b) Event accuracy reported as the mean and standard deviation across the 500 observations. Mean | Std.
---|---
0.600 | 0.0967

In order to check how well the encoder learns the encoded events, we also
report the accuracy of the event encoder in 1(b).

##  4 Discussion and Conclusion

Our results show that for around 75% of the test cases, the encoded time
series resulted in the true topology leading to the minimum parsimony. In
almost every case the encodings resulted in the second or third lowest
parsimony. This means that our approach has some potential in being able to
learn valuable binary encodings for the topology reconstruction problem.

The event accuracy is fairly low and we hypothesize that this is partly due to
a delay in the event detection. We use a causal convolutional encoder which
means that the encoder only has access to information prior to a given time.
Due to the urgency of the topology reconstruction problem not being imminent,
we believe that an ordinary convolutional encoder could lead to better results
in future work.

## References

  * De Ryck et al. [2021] T. De Ryck, M. De Vos, and A. Bertrand.  Change point detection in time series data using autoencoders with a time-invariant representation.  _IEEE Transactions on Signal Processing_ , 69:3513–3524, 2021.  10.1109/TSP.2021.3087031.  URL https://ieeexplore.ieee.org/document/9447888.
  * Fitch [1971] W. M. Fitch.  Toward defining the course of evolution: Minimum change for a specific tree topology.  _Systematic Zoology_ , 20(4):406–416, 1971.  ISSN 00397989.  URL http://www.jstor.org/stable/2412116.
  * Franceschi et al. [2019] J.-Y. Franceschi, A. Dieuleveut, and M. Jaggi.  Unsupervised scalable representation learning for multivariate time series.  In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, _Advances in Neural Information Processing Systems_ , volume 32. Curran Associates, Inc., 2019.  URL https://proceedings.neurips.cc/paper_files/paper/2019/file/53c6de78244e9f528eb3e1cda69699bb-Paper.pdf.
  * Hartigan [1973] J. A. Hartigan.  Minimum mutation fits to a given tree.  _Biometrics_ , 29(1):53–65, 1973.  ISSN 0006341X, 15410420.  URL http://www.jstor.org/stable/2529676.
  * Koch et al. [2015] G. Koch, R. Zemel, and R. Salakhutdinov.  Siamese neural networks for one-shot image recognition.  2015\.
  * Le-Khac et al. [2020] P. H. Le-Khac, G. Healy, and A. F. Smeaton.  Contrastive representation learning: A framework and review.  _IEEE Access_ , 8:193907–193934, 2020.  10.1109/ACCESS.2020.3031549.  URL https://ieeexplore.ieee.org/document/9226466.
  * Madsen [2007] H. Madsen.  _Time series analysis_.  Chapman & Hall, 2007.  ISBN 9780429195839.  10.1201/9781420059687.
  * Patterson and Rolland [2002] R. Patterson and E. Rolland.  Hybrid fiber coaxial network design.  _Operations Research_ , 50:538–551, 06 2002.  10.1287/opre.50.3.538.7737.  URL https://www.researchgate.net/publication/220244246_Hybrid_Fiber_Coaxial_Network_Design.
  * Sørensen and Pisinger [2022] S. Sørensen and D. Pisinger.  Reconstructing the tree topology in telecommunication networks.  pages 204–204, 2022.  URL https://euro2022espoo.com/.  32nd European Conference on Operational Research, EURO 2022 ; Conference date: 03-07-2022 Through 06-07-2022.

##  Appendix A Illustration of porposed approach

Our proposed approach is illustrated in Figure 1.

Figure 1: Illustration of the proposed algorithm. A Siamese network is trained
using the parsimony score on a positive (true topology) sample (green) and a
set of negative samples (red). The time series from each of the $m$ customers
are sent through the same encoder $f$ that encodes continuous time series into
an $m\times t$ matrix of binary events. Using this matrix and each of the
different topologies, the parsimony scores can be computed. Based on these the
loss is calculated and used to update the encoder.

##  Appendix B Possible tree topologies using four internal nodes

All possible topologies for a tree with four internal nodes is illustrated in
Figure 2. In this illustration we do not show the leaf nodes for simplicity.

Figure 2: All possible topologies for a network with 4 internal nodes. The
black node is the root and the colored dots are the splitters. Notice that
each of the splitters also have leaf-nodes as children that are not shown for
simplicity.

##  Appendix C Data simulation procedure

We sample each observation using the following approach.

  * •

Sampling a random topology from the 16 candidates shown in Appendix B

  * •

Sample a number of modems $M\in[50,100]$ and distribute these to the three
internal nodes (not the root)

  * •

Set $\mu_{m,t}=0$ for all modems

  * •

Divide time stamps into a set of chunks of a predertermined size $\tau$
($t_{i},t_{i+\tau}$)

  * •

For each chunk sample an internal edge on which an event occurs. Determine the
affected modems ($M^{*}$) and timepoints ($T^{*}$) and set:

$\mu_{m,t}=3,\quad m\in M^{*},t\in T^{*}$

  * •

Sample a signal using the AR(2) model:

$x_{t}=0.6x_{t-1}-0.5x_{t-2}+\epsilon_{t},\quad x_{0}=x_{1}=0$

where $\epsilon_{t}\sim\mathcal{N}(\mu_{t},0.5^{2})$


# Tuning electronic and magnetic properties of FeRh alloy by chemical and
physical method

Greeshma R  Rudra Banerjee<EMAIL_ADDRESS>Department of Physics and
Nanotechnology, SRM Institute of Science and Technology, Kattankulathur, Tamil
Nadu, 603203, India

###### Abstract

The electronic, magnetic, and thermodynamic properties of ordered and
chemically disordered FeRh alloy is studied using ab-initio methods. The
equiatomic Fe50Rh50 composition is reported for both ordered and disordered
phases. Chemically disordered FexRh100-x is reported and the effect of
disorder on electronic and magnetic properties is discussed. Further, We have
reported the effects of stress and strain in both the order and disorder
phases. The result is only for the cubic phase and no distortion has been
taken into consideration. This study is motivated by the recent resurgence in
FeRh study motivated by the fact that the barocaloric properties can be
possible to sustain over the cycle. Hence, we have discussed the properties of
FexRh100-x with chemical disorder and pressure together, to gain an insight
into the compound effect and the interplay between them.

##  I Introduction

The century old Joule-Thomson effect for cooling technology is not only
inefficient but also emits environmentally hazardous by-products. In the last
decade caloric materials(CM) has been staged as a feasible alternative due to
its environment friendly cooling technology and improved efficiency[1]. The
caloric materials show significant adiabatic temperature change ($\Delta T$)
or isothermal entropy change ($\Delta S$) near phase boundary under the
applied magnetic, electric or mechanical field, corresponding to
magnetocaloric(MC)[2], electrocaloric(EC) or mechanocaloric(mC) elasto(eC) or
barocaloric(BC)) effects[3]. Though the magnetocaloric effect is believed to
be observed first in 19th century[4], the field really took off with the
observation of ‘giant’ mC[5], MC[6] and EC[7] in 1980s and subsequent decades.

Various CMs, particularly based on Lanthanides(specially, Gd)[8] and
transition metal (specially, Mn[9] and Fe) materials have shown impressive
caloric properties. The multicaloric materials, where materials can show
caloric properties stimulated by one or multiple fields as described above,
are rare and far between. While there are theoretical understanding of
multicaloric systems[10], experiments are mostly constrained by individual
fields[11]. Off-stoichiometric FeRh system is one of the known systems showing
inverse-MC(iMC), BC and EC. FeRh shows giant iMC near room temperature[6] at
$350K$ from a first-order antiferromagnetic (AFM) to ferromagnetic (FM)
transition. The transition results around 1% volume expansion, leading to the
BC property of FeRh[12, 13, 14, 15]. FeRh exhibits around $\Delta S\approx
20JK^{-1}kg^{-1}$ and $\Delta T\approx 20K$ at 2T, making it an excellent CM.
But, FeRh also shows broad hysteresis[16, 13], leading to a degradation of
caloric properties over refrigeration cycles. However, Liu et.al has shown[17]
that it is possible to enhance the property over the cycles. Qiao et.al[18]
has recently shown the effect of growth on MC properties of FeRh.

This renewed interest in off-stoichiometric FeRh system demands a detailed
look into the electronic and magnetic structure of FexRh100-x, where $x$ is
the percentage of concentration at a site ranging from 49-51, with varying
pressure. In this article, we have delineated our systematic study of
FexRh100-x, subject to hydrostatic pressure. We have studied the variation in
the electronic structure, magnetic exchange and magnetic transition
temperature of FeRh alloy. We carried out the study of this alloy in both FM
and AF domain. The study unveiled that the ground state FexRh100-x system is
ferromagnetic in nature(Fig. (1(c))). Hence, in this report, we have focused
the complete study on FM behaviour of FexRh100-x.

##  II Methods

The off-stoichiometric composition and disordered alloy system is best managed
using multiple scattering Green’s function and Coherent Potential
Approximation (CPA) respectively. The electronic and magnetic calculations in
this study are performed using SPRKKR code[19, 20]. The Perdew, Burke and
Ernzerhoff(PBE) generalized gradient approximation (PBE-GGA) is used for
exchange-correlational functional[21]. We performed first Brillouin zone
integrations on the 2500 grid of $k$-points and the energy convergence
criteria had been set to $10^{-5}Ry$. The full potential spin polarised scalar
relativistic calculations with angular momentum cut-off $l_{max}=2$ has been
performed.

Initially we determined the lattice parameter with minimum energy for the
Fe50Rh50 structure using the following procedure; i) Procured the lattice
parameter from materials project database. ii) Computed the scf of the system,
by varying lattice parameter ranging from 94% to 106%, similar calculation is
done for each lattice parameters. iii) The curve of lattice parameter vs
energy is fitted by quadratic polynomial equation. iv) The minimum of the
curve is the optimised lattice parameter. For off-stoichiometric structures,
we took the optimised lattice parameter from the stoichiometric calculations
as the starting point and followed the steps mentioned above. The optimisation
curve of ordered Fe50Rh50 is shown in the Fig. (1(b)).

The magnetic exchange energy ($\mathcal{J}_{ij}$) was evaluated to understand
the effects of magnetic interactions using the Lichtenstein formula[22, 23].

$\displaystyle\mathcal{J}_{ij}$ $\displaystyle=\frac{1}{4\pi}\int d\varepsilon
f(\beta(\varepsilon-\varepsilon_{F}))ImTr[\hat{\Delta}_{i}\hat{T}^{ij}_{\uparrow}\hat{\Delta}_{j}\hat{T}^{ji}_{\downarrow}]$
(1) $\displaystyle=\frac{1}{4\pi}\int d\varepsilon
f(\beta(\varepsilon-\varepsilon_{F}))ImTr[\hat{G}^{+}_{\uparrow}(\varepsilon)\hat{P}_{i}\hat{G}^{+}_{\downarrow}(\varepsilon)\hat{P}_{j}]$

where, $\hat{T}^{ij}_{\sigma}$ is the scattering path operator for sites $i$
and $j$, $\hat{\Delta}\equiv\hat{t}_{i\uparrow}-\hat{t}_{j\downarrow}$ is the
single site scattering matrix $\hat{t}_{i\sigma}$ at site $i$,
$f(x)=1/(e^{x}+1)$, $\beta=1/k_{B}T$ and $\varepsilon_{F}$ is the Fermi
energy, $\hat{G}^{+}_{\sigma}(\varepsilon)$ is the retarded Green’s function
of the spin $\sigma$ in unperturbed state and
$\hat{P}_{i}\equiv\hat{H}_{i\uparrow}-\hat{H}_{i\downarrow}$ where,
$\hat{H}_{i\sigma}$ is the on-site term of the Hamiltonian for spin $\sigma$
at site $i$. The $\mathcal{J}_{ij}$ is reckoned by the energy difference due
to the infinitesimal change in magnetic direction, as mapped out from Eqn(1).

(a) FeRh unit cell with CsCl structure.

(b) Calculation of minimum energy Lattice parameter for Fe50Rh50.

(c) Energy difference of FexRh100-xin FM and AFM magnetic structure.

Figure 1: Initial information of FexRh100-x. (1(a)) shows the FexRh100-x
structure. In ordered structure Fe and Rh occupies either corner (0,0,0) or
central (.5,.5,.5) site. In disordered systems both (0,0,0) and (.5,.5,.5)
sites are occupied by Fe and Rh as per concentration. (1(b)) shows the energy
minimisation of lattice structure to find the ground state lattice parameter.
(1(c)) shows the energy difference between AFM and FM state. In the $y$-axis,
$\Delta E=E_{mag}-E_{FM}$ is plotted, where $E_{mag}$ is either $E_{FM}$ or
$E_{AFM}$.

Finally, the Curie temperature is computed using Mean Field Theory(MFT):

$k_{B}T_{C}=\frac{2}{3}\mathcal{J}_{ij}$ (2)

where, $\mathcal{J}_{ij}$ is the largest eigen value of the determinant[24,
25].

##  III Result

FeRh has CsCl structure in ordered phase(B2-phase), with space group 221
(Pm$\bar{3}$m), as shown in Fig. (1(a)). In the ordered phase, Fe and Rh
occupies $(0,0,0)$ and $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$
sublattice, respectively. In chemically disordered phase, Fe50Rh50 stabilises
is bcc-A2 structure[25]. In this phase, both Fe and Rh is occupying both
sublattices with equal probability. Fe51Rh49 and Fe49Rh51 systems are also
disordered with (0,0,0) site is Fe rich or deficient, respectively, as
described in Fig. (1(a)).

###  III.1 Ordered FeRh system

####  III.1.1 Electronic Structure

The total and atoms-projected density of states (DOS) for this system is shown
in the Fig. (2). Fig. (2(b)) shows the DOS of Fe50Rh50 at optimised lattice
parameter.

(a) DOS for 6% strained lattice parameter($a$).

(b) DOS for optimised lattice parameter($a$).

(c) DOS for 6% stretched lattice parameter($a$).

Figure 2: The electronic structure of ordered Fe50Rh50.

Fig. (2(a)) is DOS of Fe50Rh50 with 6% compressed lattice parameter,
resembling very high hydrostatic pressure. Likewise, Fig. (2(c)) shows the
electronic structure of Fe50Rh50 with 6% stretch.

The Fig. (2) shows the DOS of Fe50Rh50 with (2(a))strained, (2(b))optimised
and (2(c))stretched lattice parameter. The pseudogap around 1eV down in
minority spin channel in optimised lattice structure is originated from the
$3d$ hybridisation of Fe and Rh. The peak just below the $E_{F}$ in Fig.
(2(a)) in the majority spin channel, originating from the hybridisation of the
same orbitals shows the Jahn-Teller instability[26, 27]. With the increase in
lattice parameter $a$, the peak moves further down from $E_{F}$, stabilising
against the possible Jahn-Teller distortion. The volume-preserving
stretch/strain with varying $c/a$ ratio may show the enhancement of Jahn-
Teller distortion, but the possibility within cubic phase decreases with
increase in $a$.

In real systems, such a high hydrostatic pressure or stretch is not only
nonphysical, but may have other unforeseen effects as well. Here we have used
this to check the extreme conditions. In all the following discussions, this
6% stretch/strain has been considered without further explanation.

####  III.1.2 Magnetic Properties

Element resolved magnetic moments $\mu_{Fe},\mu_{Rh}$ and $\mu_{FeRh}$ of
ordered Fe50Rh50 is shown in Table (1). For optimised and strained structure,
$\mu_{Fe}>3\mu_{B}$ as already noted by previous study[28].

(a) $\mathcal{J}_{ij}$ for 6% strained lattice parameter($a$).

(b) $\mathcal{J}_{ij}$ for optimised lattice parameter($a$).

(c) $\mathcal{J}_{ij}$ for 6% stretched lattice parameter($a$).

Figure 3: Magnetic interaction ($\mathcal{J}_{ij}$) of ordered Fe50Rh50.

Fig. (3) shows the magnetic interactions of ordered Fe50Rh50 with Fe of
(0,0,0) sublattice at the center. The stretched configuration in Fig. (3(a))
is interesting as it is showing very strong alternating positive and negative
interaction, leading to a spin-glass like configuration. The glassy phase in
FeRh is already known for metastable fcc phase[29, 30, 31], but to the best of
our knowledge, we are the first to observe spin-glass effect in these 221
systems using pressure only. In optimised or stretched system, the glassy
behaviour is not observed, and dominant interactions are ferromagnetic.

###  III.2 Disordered FeRh system

a.1 DOS for 6% strained lattice parameter($a$).

a.2 DOS for optimised lattice parameter($a$).

a.3 DOS for 6% stretched lattice parameter($a$).

(a) The DOS of disordered Fe49Rh51.

b.1 DOS for 6% strained lattice parameter($a$).

b.2 DOS for optimised lattice parameter($a$).

b.3 DOS for 6% stretched lattice parameter($a$).

(b) The DOS of disordered Fe50Rh50.

c.1 DOS for 6% strained lattice parameter($a$).

c.2 DOS for optimised lattice parameter($a$).

c.3 DOS for 6% stretched lattice parameter($a$).

(c) The DOS of disordered Fe51Rh49.

Figure 5: Electronic structure of disordered FexRh100-x systems for
(5(a))$x=$49,(5(b))50, and (5(c))51. While the electronic structures does not
change significantly with $x$, it does changes with lattice parameter $a$: (a)
The width of the DOS squeezes with increasing $a$. (b) The spin-splitting in
atom resolved DOS increases with $a$.

In disordered systems Fe50Rh50 shows bcc-A2 structure. In this case, Fe and Rh
occupies both (0,0,0) and (.5,.5,.5) sites with probability according to their
concentrations. For brevity, we will denote Fe/Rh at (0,0,0) as Fe1/Rh1 and
Fe/Rh at (.5,.5,.5) as Fe2/Rh2. It has to be noted that in this phase, there
are 4 atoms in unit cell, unlike 2 atoms in section (III.1)

####  III.2.1 Electronic structure

The DOS for disordered phases are shown in the Fig. (5). The DOS is much
smeared w.r.t. the ordered DOS plotted in Fig. (2), which is a characteristic
of disordered systems[28, 32]. Still, the main features of ordered state is
still visible in disordered state as well. The valley near the -2eV in upspin
channel is the only distinct feature in strained state. The Rh DOS does not
change significantly over the lattice parameter ranges studied, the Fe atom
starts showing more and more structures, bringing more features in total DOS.
Beside the deep valley, there is also a saddle neck just below the $E_{F}$ in
major spin channel gets more and more visible. Likewise, the pseudo-gap
discussed in the section (III.1.1) appears in both optimised and strained
state.

####  III.2.2 Magnetic Properties

Similar to the electronic structures, magnetic interactions with Fe atom at
the center does not change much with concentrations. Unlike the ordered phase
in section (III.1.2), there is no visible spin glass phase in disordered
phase.

a.1 $\mathcal{J}_{ij}$ for 6% strained lattice parameter($a$).

a.2 $\mathcal{J}_{ij}$ for optimised lattice parameter($a$).

a.3 $\mathcal{J}_{ij}$ for 6% stretched lattice parameter($a$).

(a) The DOS and $\mathcal{J}_{ij}$ of disordered Fe49Rh51.

b.1 $\mathcal{J}_{ij}$ for 6% strained lattice parameter($a$).

b.2 $\mathcal{J}_{ij}$ for optimised lattice pa rameter($a$).

b.3 $\mathcal{J}_{ij}$ for 6% stretched lattice parameter($a$).

(b) The $\mathcal{J}_{ij}$ of disordered Fe50Rh50.

c.1 $\mathcal{J}_{ij}$ for 6% strained lattice parameter($a$).

c.2 $\mathcal{J}_{ij}$ for optimised lattice parameter($a$).

c.3 $\mathcal{J}_{ij}$ for 6% stretched lattice parameter($a$).

(c) The $\mathcal{J}_{ij}$ of disordered Fe51Rh49.

Figure 7: $\mathcal{J}_{ij}$ of disordered FexRh100-x systems for
(7(a))$x=$49, (7(b))50 and (7(c))51 with Fe1 at the center. Similar to the
electronic strictures, magnetic exchange interactions is not much changed with
variations in $x$. The stress/strain, i.e. variation in $a$ changes the
$\mathcal{J}_{ij}$ significantly. The $\mathcal{J}_{0}^{Fe}$ decreases
slightly with increasing $a$, but $\mathcal{J}_{0}^{Rh}$ has increased to
almost double with 12% change in $a$.

The variation in magnetic moments of FexRh100-x and Fe atoms are shown in Fig.
(8(a)) and Fig. (8(b)). The values are tabulated in last three rows of the
table (1). While the moment of Fe is very close to their atomic level (2.412
$\mu_{B}$ in 221 structure[33]), the moment increases monotonously with the
lattice parameter. This trend shows magnetic properties are more dependent on
the distance between the atoms rather than the concentrations in this off-
stoichiometric systems. The magnetic moment of both Fe and Rh increase with
the lattice parameter. This is due to the increased spin-polarisation in both
Fe and Rh, as seen in atom projected DOS in Fig. (5). This behaviour is
completely different from volume-preserved distortion, where $\mu_{Rh}$,
$\mu_{Fe}$, $\mathcal{J}_{0}^{Rh}$ and $\mathcal{J}_{0}^{Fe}$ decreases
lattice parameter[25]. This indicates an establishment of long-range Fe-Fe FM
interaction in this case, instead of a competing short-range FM and long-range
AFM interactions between Fe.

(a) Magnetic moment of FexRh100-x.

(b) Magnetic moment of Fe atom in FexRh100-x.

(c) $T_{C}$ of FexRh100-x.

Figure 8: Variation of Magnetic moment and Curie temperature of ordered
FexRh100-x with $x$ and $a$.

The Curie temperature calculated using MFT(Eqn(2)) is tabulated in the table
(1) and Fig. (8(c)). Decreasing the Fe content also decreases the $T_{C}$. Our
MFT result for B2-Fe50Rh50 matches earlier results obtained using same method.
The lattice parameter effects $T_{C}$ profoundly, with around 70% increase in
$T_{C}$ for 12% change in lattice parameter. The higher magnetic splitting in
Rh has increased the inter-sublattice interaction, leading to this high
$\mathcal{J}_{ij}$, leading to the higher $T_{C}$.

Fe | Rh |  | Magnetic moment ($\mu_{B}$) | $T_{C}(K)$
---|---|---|---|---
Fe | Rh | FeRh
50o | 50 | Stretched | 2.9317 | 0.9319 | 3.8636 | 539
50 | Optimised | 3.2711 | 0.9417 | 4.2128 | 820
50 | Strained | 3.3929 | 0.9340 | 4.3269 | 832
49 | 51 | Stretched | 2.7786 | 0.4301 | 3.1616 | 522
51 | Optimised | 3.0774 | 0.7173 | 3.7475 | 772
51 | Strained | 3.2200 | 0.9620 | 4.1369 | 907
50d | 50 | Stretched | 2.7472 | 0.4239 | 3.1711 | 527
50 | Optimised | 3.0703 | 0.7253 | 3.7956 | 760
50 | Strained | 3.2062 | 0.9480 | 4.1542 | 908
51 | 49 | Stretched | 2.7679 | 0.4494 | 3.2637 | 546
49 | Optimised | 3.0631 | 0.7331 | 3.8427 | 784
49 | Strained | 3.2058 | 0.9531 | 4.2042 | 912
Table 1: Magnetic properties of FexRh100-x. The 50o and 50d in first and third
row means the results are for ordered and disordered Fe50Rh50.

##  IV Conclusion

In this communication, we have presented, electronic and magnetic properties
of FexRh100-x as obtained from density functional theory based calculations.
The main outcome of the calculations can be summarised as  (i) The electronic
structures of FexRh100-x matches well with previous calculations.  (ii) The
magnetic moment of both Fe and Rh is higher than the free atoms. The magnetic
moment of Rh in disordered phase is due to the hybridisation of nearest
neighbour Fe.  (iii) Both electronic and magnetic properties has stronger
dependency on the distance between nearest neighbours or the lattice
parameters than the small variations in composition.  Particularly, we have
shown the effect of chemical disorder and pressure on electronic and magnetic
properties of FexRh100-xboth separately and together. This will help further
understanding and designing Fe-based caloric materials.

##  V Acknowledgment

We acknowledge the High Performance Computing Center (HPCC), SRM IST for
providing the computational facility to carry out this research work
successfully.

## References

  * [1] L. J. Swartzendruber.  The fe-rh (iron-rhodium) system.  Bulletin of Alloy Phase Diagrams, 5(5):456–462, oct 1984.
  * [2] J. B. Staunton, R. Banerjee, M. dos Santos Dias, A. Deak, and L. Szunyogh.  Fluctuating local moments, itinerant electrons, and the magnetocaloric effect: Compositional hypersensitivity of FeRh.  Physical Review B, 89(5), feb 2014.
  * [3] X. Moya, S. Kar-Narayan, and N. D. Mathur.  Caloric materials near ferroic phase transitions.  Nature Materials, 13(5):439–450, apr 2014.
  * [4] Anders Smith.  Who discovered the magnetocaloric effect?  The European Physical Journal H, 38(4):507–517, jun 2013.
  * [5] C. Rodriguez and L. C. Brown.  The thermal effect due to stress-induced martensite formation in $\beta$-CuAlNi single crystals.  Metallurgical Transactions A, 11(1):147–150, dec 1980.
  * [6] S.A. Nikitin, G. Myalikgulyev, A.M. Tishin, M.P. Annaorazov, K.A. Asatryan, and A.L. Tyurin.  The magnetocaloric effect in fe49rh51 compound.  Physics Letters A, 148(6-7):363–366, aug 1990.
  * [7] A. S. Mischenko, Q. Zhang, J. F. Scott, R. W. Whatmore, and N. D. Mathur.  Giant electrocaloric effect in thin-film PbZr0.95Ti0.05O3.  Science, 311(5765):1270–1271, mar 2006.
  * [8] A. Fujita, S. Fujieda, Y. Hasegawa, and K. Fukamichi.  Itinerant-electron metamagnetic transition and large magnetocaloric effects in $\mathrm{La}({\mathrm{fe}}_{x}{\mathrm{si}}_{1-x}{)}_{13}$ compounds and their hydrides.  Physical Review B, 67(10), mar 2003.
  * [9] Nguyen H. Dung, Zhi Qiang Ou, Luana Caron, Lian Zhang, Dinh T. Cam Thanh, Gilles A. de Wijs, Rob A. de Groot, K. H. Jürgen Buschow, and Ekkes Brück.  Mixed magnetism for refrigeration and energy conversion.  Advanced Energy Materials, 1(6):1215–1219, sep 2011.
  * [10] Yang Liu, Ingrid C Infante, Xiaojie Lou, Laurent Bellaiche, James F Scott, and Brahim Dkhil.  Giant room-temperature elastocaloric effect in ferroelectric ultrathin films.  Advanced Materials, 26(35):6132–6137, 2014.
  * [11] J Krishna Murthy and A Venimadhav.  Multicaloric effect in multiferroic y2comno6.  Journal of Physics D: Applied Physics, 47(44):445002, 2014.
  * [12] Enric Stern-Taulats, Antoni Planes, Pol Lloveras, Maria Barrio, Josep-Lluí s Tamarit, Sabyasachi Pramanick, Subham Majumdar, Carlos Frontera, and Lluís Mañosa.  Barocaloric and magnetocaloric effects in Fe49Rh51.  Physical Review B, 89(21), jun 2014.
  * [13] Enric Stern-Taulats, Adrià Gràcia-Condal, Antoni Planes, Pol Lloveras, Maria Barrio, Josep-Lluís Tamarit, Sabyasachi Pramanick, Subham Majumdar, and Lluís Mañosa.  Reversible adiabatic temperature changes at the magnetocaloric and barocaloric effects in fe49rh51.  Applied Physics Letters, 107(15):152409, 2015.
  * [14] V. L. Moruzzi and P. M. Marcus.  Antiferromagnetic-ferromagnetic transition in ferh.  Phys. Rev. B, 46(0):2864–2873, Aug 1992.
  * [15] Enric Stern-Taulats, Antoni Planes, Pol Lloveras, Maria Barrio, Josep-Lluís Tamarit, Sabyasachi Pramanick, Subham Majumdar, Carlos Frontera, and Lluís Mañosa.  Barocaloric and magnetocaloric effects in ${\mathrm{fe}}_{49}{\mathrm{rh}}_{51}$.  Phys. Rev. B, 89(8):214105, Jun 2014.
  * [16] V Franco, JS Blázquez, B Ingale, and A Conde.  The magnetocaloric effect and magnetic refrigeration near room temperature: materials and models.  Annual Review of Materials Research, 42:305–342, 2012.
  * [17] Yang Liu, Lee C. Phillips, Richard Mattana, Manuel Bibes, Agnès Barthélémy, and Brahim Dkhil.  Large reversible caloric effect in FeRh thin films via a dual-stimulus multicaloric cycle.  Nature Communications, 7(1), may 2016.
  * [18] Kaiming Qiao, Yuhang Liang, Hu Zhang, Fengxia Hu, Ziyuan Yu, Yi Long, Jing Wang, Jirong Sun, Tongyun Zhao, and Baogen Shen.  Manipulation of magnetocaloric effect in FeRh films by epitaxial growth.  Journal of Alloys and Compounds, 907:164574, jun 2022.
  * [19] H Ebert and et.al.  The munich spr-kkr package.
  * [20] H Ebert, D Ködderitzsch, and J Minár.  Calculating condensed matter properties using the KKR-green's function method—recent developments and applications.  Reports on Progress in Physics, 74(9):096501, aug 2011.
  * [21] John P. Perdew, Kieron Burke, and Matthias Ernzerhof.  Generalized gradient approximation made simple.  Phys. Rev. Lett., 77:3865–3868, Oct 1996.
  * [22] A I Liechtenstein, M I Katsnelson, and V A Gubanov.  Exchange interactions and spin-wave stiffness in ferromagnetic metals.  Journal of Physics F: Metal Physics, 14(7):L125–L128, jul 1984\.
  * [23] Asako Terasawa, Munehisa Matsumoto, Taisuke Ozaki, and Yoshihiro Gohda.  Efficient algorithm based on liechtenstein method for computing exchange coupling constants using localized basis set.  Journal of the Physical Society of Japan, 88(11):114706, nov 2019\.
  * [24] A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov, and V.A. Gubanov.  Local spin density functional approach to the theory of exchange interactions in ferromagnetic metals and alloys.  Journal of Magnetism and Magnetic Materials, 67(1):65–74, may 1987\.
  * [25] Ralf Witte, Robert Kruk, Markus E. Gruner, Richard A. Brand, Di Wang, Sabine Schlabach, Andre Beck, Virgil Provenzano, Rossitza Pentcheva, Heiko Wende, and Horst Hahn.  Tailoring magnetic frustration in strained epitaxial FeRh films.  Physical Review B, 93(10), mar 2016.
  * [26] Tufan Roy, Dhanshree Pandey, and Aparna Chakrabarti.  Probing the possibility of coexistence of martensite transition and half-metallicity in ni and co-based full-heusler alloys: An$abinitio$calculation.  Physical Review B, 93(18), may 2016.
  * [27] C. Bordel, J. Juraszek, David W. Cooke, C. Baldasseroni, S. Mankovsky, J. Minár, H. Ebert, S. Moyerman, E. E. Fullerton, and F. Hellman.  Fe spin reorientation across the metamagnetic transition in strained ferh thin films.  Phys. Rev. Lett., 109:117201, Sep 2012.
  * [28] J. Kudrnovský, V. Drchal, and I. Turek.  Physical properties of FeRh alloys: The antiferromagnetic to ferromagnetic transition.  Physical Review B, 91(1), jan 2015.
  * [29] E Navarro, M Multigner, A. R Yavari, and A Hernando.  The spin glass state of metastable fcc FeRh.  Europhysics Letters (EPL), 35(4):307–312, aug 1996.
  * [30] Jiahui Chen, Ya Gao, Liang Wu, Jing Ma, and Ce-Wen Nan.  A magnetic glass state over the first-order ferromagnetic-to-antiferromagnetic transition in FeRh film.  Materials Research Letters, 5(5):329–334, feb 2017.
  * [31] Radhika Barua, Xiujuan Jiang, Felix Jimenez-Villacorta, J. E. Shield, D. Heiman, and L. H. Lewis.  Tuning the magnetostructural phase transition in FeRh nanocomposites.  Journal of Applied Physics, 113(2):023910, jan 2013.
  * [32] Rudra Banerjee and Abhijit Mookerjee.  Augmented space recursion code and application in simple binary metallic alloy.  International Journal of Modern Physics C, 21(02):205–220, feb 2010\.
  * [33] Materials Project database.


# Embedding of Higson compactification into the product of adelic solenoids.

Alexander Dranishnikov  and  James Keesling  Alexander N. Dranishnikov,
Department of Mathematics, University of Florida, 358 Little Hall,
Gainesville, FL 32611-8105, USA<EMAIL_ADDRESS>James E. Keesling, Department
of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL
32611-8105, USA<EMAIL_ADDRESS>
###### Abstract.

The Higson compactification of any simply connected proper geodesic metric
space admits an embedding into a product of adelic solenoids that induces an
isomorphism of 1-dimensional cohomology.

###### 2000 Mathematics Subject Classification:

Primary 53C23, 54F15; Secondary 22C05, 54F15, 55N05

The first author was supported by Simons Foundation

##  1\. Introduction

The Higson corona of a metric space was introduced by Higson and Roe
[Ro1],[Ro2] to bring an approximation of the $K$-theory of its Roe algebra
$C_{roe}^{*}(X)$ by the topological $K$-theory of the corona. The $K$-theory
of the Roe algebra is the recipient in the coarse assembly map and therefore,
it plays an important role in a coarse approach to the Novikov and the Baum-
Connes conjectures. It turns out that for injectivity of the coarse assembly
map it is sufficient to have the Higson compactification acyclic [Ro1]. James
Keesling noticed that the Higson compactification of $\mathbb{R}^{n}$ is not
acyclic in dimension one [K]. Later Dranishnikov and Ferry have constructed
nontrivial second cohomology classes of the Higson compactification of
$\mathbb{R}^{2}$ [DF].

It turns out that all the above cohomology classes of the Higson
compactifications are divisible. This leaves a hope that all cohomology groups
of the Higson compactification are vector spaces over $\mathbb{Q}$. In that
case the Higson compactification would be acyclic with respect to any finite
coefficient group. In [DFW] Dranishnikov, Ferry, and Weinberger proved the
Novikov Higher Signature conjecture for manifolds whose fundamental group
$\Gamma$ admits a finite classifying space $B\Gamma$ with trivial Čech
cohomology of the Higson compactification $\overline{E\Gamma}$ of the
universal cover $E\Gamma$ with $\mathbb{Z}_{2}$ coefficients. In the same
paper it was proven that $\check{H}^{*}(\overline{E\Gamma};\mathbb{Z}_{p})=0$
for all prime $p$ whenever the asymptotic dimension $\operatorname{{\rm
asdim}}\Gamma$ of the group $\Gamma$ is finite.

In this paper we construct an embedding of $\overline{E\Gamma}$ into the
product of adelic solenoids $\prod\Sigma_{p}$ which induces isomorphism of
rational cohomology. In particular, our embedding implies that
$\check{H}^{1}(\overline{E\Gamma})=\oplus\mathbb{Q}$ and, hence,
$\check{H}^{1}(\overline{E\Gamma};\mathbb{Z}_{p})=0$ for all $p$.

We think that it is a reasonable to believe that our embedding induces an
epimorphism of cohomology groups for higher dimensions. This conjecture seems
plausible in the case when $\dim\overline{E\Gamma}<\infty$. We note that from
[DFW] one can derive our conjecture for groups with $\operatorname{{\rm
asdim}}\Gamma<\infty$. We recall that always
$\dim\nu(E\Gamma)\leq\operatorname{{\rm asdim}}\Gamma$ [DKU] and
$\dim\nu(E\Gamma)=\operatorname{{\rm asdim}}\Gamma$ [Dr] when the latter is
finite. Here $\nu(E\Gamma)=\nu\Gamma$ is the Higson corona:
$\nu(E\Gamma)=\overline{E\Gamma}\setminus E\Gamma$. There is a long open
question whether there exists a metric spaces $X$ with $\dim\nu X<\infty$ and
$\operatorname{{\rm asdim}}\Gamma=\infty$. Perhaps a more challenging related
question here is whether the Novikov conjecture holds true for groups with
$\dim\nu\Gamma<\infty$. We recall that the Novikov conjecture for groups with
$\operatorname{{\rm asdim}}\Gamma<\infty$ was proven more than two decades ago
in [Yu].

##  2\. Preliminaries

###  2.1. Higson compactification

Let $(X,d)$ be a metric space with a base point $x_{0}$. By $B(r,x)$ we denote
the open ball of radius $r$ with the center in $x$. We call a map $f:X\to Y$
to a metric space $Y$ slowly oscillating if for any $r>0$ the diameter of
$f(B(x,r))$ tends to 0 as $d(x,x_{0})$ tends to infinity. The Higson
compactification $\bar{X}$ of $X$ is characterized by bounded slowly
oscillating continuous functions $f:X\to\mathbb{R}$. If $C_{h}$ is the ring of
all such functions, then $\bar{X}$ is homeomorphic to the closure of $X$ under
the embedding

$(f)_{f\in C_{h}}:X\to\prod_{f\in C_{h}}[\inf f,\sup f].$

We use notation $\nu X=\bar{X}\setminus X$ for the Higson corona. The
following is straightforward.

######  2.1 Proposition.

For a metric space $X$ a continuous map $f:X\to C$ to a compact metric space
$C$ extends continuously to the Higson corona if and only if $f$ is slowly
oscillating.

A metric space $X$ is called proper if the distance to each point in $X$ is a
proper function. A metric space $X$ is called geodesic if for any pair of
points $x,y\in X$ there is an isometric embedding $\xi:[0,d(x,y)]\to X$ with
$\xi(0)=x$ and $\xi(d(x,y))=y$.

The following can be found in [DKU].

######  2.2 Proposition.

In a proper metric space $X$ closed subsets $Y$ and $Y^{\prime}$ have common
points in the Higson corona, $Cl_{\bar{X}}Y)\cap
Cl_{\bar{X}}(Y^{\prime})\cap\nu X\neq\emptyset$, if and only if $d(Y\setminus
B(x_{0},r),Y^{\prime})\to\infty$ as $r\to\infty$.

###  2.2. Solenoids

We start from the definition of the $p$-adic solenoid $\Sigma_{p}$ as the
inverse limit:

$\Sigma_{p}=\lim_{\leftarrow}\\{S^{1}\stackrel{{\scriptstyle
p}}{{\leftarrow}}S^{1}\stackrel{{\scriptstyle
p}}{{\leftarrow}}S^{1}\leftarrow\cdots\\}$

of unit circles $S^{1}\subset\mathbb{C}$ where the bonding maps are group
homomorphisms defined by the map $z\mapsto z^{p}$. Thus, $\Sigma_{p}$ is an
abelian topological group. We use $+$ for the group operation in $\Sigma_{p}$.
Let $p_{0}:\Sigma_{p}\to S^{1}$ be the projection onto the first circle in the
inverse sequence. Note that the subgroup
$p_{0}^{-1}(1)=A_{p}\subset\Sigma_{p}$ is the group of $p$-adic integers. We
use the notation $A_{p}$ for $p$-adic integers instead of more common notation
from algebra $\mathbb{Z}_{p}$, since the later is used in topology (and
already was used in this paper) for the $\mathbb{Z}$ $mod\ p$ group.

Let $u:\mathbb{R}\to S^{1}\subset\mathbb{C}$ be the universal cover given by
the exponential map $u(t)=e^{2\pi it}$. Thus, $u$ is a group homomorphism with
$u^{-1}(1)=\mathbb{Z}$. Then the lift $\tilde{u}:\mathbb{R}\to\Sigma_{p}$ that
takes $0$ to the unit of $\Sigma_{p}$ is an injective group homomorphism.
Thus, the group of reals is a subgroup of the solenoid,
$\mathbb{R}\subset\Sigma_{p}$. We note that $\mathbb{R}\cap A_{p}=\mathbb{Z}$.
Since $\mathbb{Z}$ is dense in $A_{p}$, it follows that $\mathbb{R}$ is dense
in $\Sigma_{p}$. Also we note that all path components of $\Sigma_{p}$ are the
$\mathbb{R}$-cosets $a+\mathbb{R}$, $a\in\Sigma_{p}$. They are the bijective
images of the reals $g:\mathbb{R}\to\Sigma_{p}$ that are obtained as different
lifts of the universal cover $u$ with respect to $p_{0}$. Another way to
define the $p$-adic solenoid is taking the qoutient group
$\Sigma_{p}=(\mathbb{R}\times A_{p})/\mathbb{Z}$ for the natural diagonal
embedding $\mathbb{Z}\to\mathbb{R}\times\mathbb{A}_{p}$.

The adelic solenoid is the quotient
$\Sigma_{\mathbb{Q}}=\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ of the adels of the
rationals [BV]. It can be viewed as the quotient group
$\Sigma_{\mathbb{Q}}=(\mathbb{R}\times\hat{\mathbb{Z}})/\mathbb{Z}$ where
$\hat{\mathbb{Z}}$ is the profinite completion of the integers and
$\mathbb{Z}\to\mathbb{R}\times\hat{\mathbb{Z}}$ is the diagonal map for
natural inclusions $\mathbb{Z}\subset\mathbb{R}$ and
$\mathbb{Z}\subset\hat{\mathbb{Z}}$. We note that the image of the inclusion
$\mathbb{Z}\subset\hat{\mathbb{Z}}$ is dense as well as the image of the
inclusion $\mathbb{R}\subset\Sigma_{\mathbb{Q}}$. The adelic solenoid admits
the following description as the inerse limit

$\Sigma_{\mathbb{Q}}=\lim_{\leftarrow}\\{S^{1}\stackrel{{\scriptstyle
z^{2}}}{{\leftarrow}}S^{1}\stackrel{{\scriptstyle
z^{3}}}{{\leftarrow}}S^{1}\stackrel{{\scriptstyle
z^{4}}}{{\leftarrow}}\cdots\\}$

where the bonding homomorphism from the $n$th circle to the previous is taking
to the power $n$, $z\mapsto z^{n}$. Again we denote by
$p_{0}:\Sigma_{\mathbb{Q}}\to S^{1}$ the projection to the first circle. All
the above facts about $\Sigma_{p}$ hold true for $\Sigma_{\mathbb{Q}}$ with
replacement of $p$-adic integers by the profinite completion of $\mathbb{Z}$.
In this paper we will be using the notations $\Sigma_{\infty}$ for
$\Sigma_{\mathbb{Q}}$ and $A_{\infty}$ for $\hat{\mathbb{Z}}$.

######  2.3 Proposition.

For the Higson compactification $\bar{X}$ of a simply connected geodesic
metric space $X$ there is an isomorphism

$\check{H}^{1}(\bar{X})=\bigoplus_{\mathcal{A}}\mathbb{Q}.$

###### Proof.

The idea of the proof is taken from [K]. It suffices to show that each element
$\alpha\in\check{H}^{1}(\bar{X})$ is $p$-divisible for all $p$. Let
$\phi:\bar{X}\to S^{1}$ be a map representing $\alpha$, $\alpha=\phi^{*}(1)$,
where $1\in\mathbb{Z}=H^{1}(S^{1})$. Since $X$ is simply connected, there is a
lift $f:X\to\Sigma_{p}$ with respect to $p_{0}$. We fix a metric on
$\Sigma_{p}$ and define $\epsilon(d)$ to be the maximal diameter of a
component of $p_{0}^{-1}(B(s,d))$ for $s\in S^{1}$ and the standard metric on
$S^{1}$. Clearly, $\lim_{d\to 0}\epsilon(d)=0$. Since the balls $B(x,r)$ in a
geodesic metric space are connected, we obtain

$diamf(B(x,r))\leq\epsilon(diam(\phi(B(x,r))).$

Then the slow oscillation of $\phi$ implies that $f$ is slow oscillating.
Hence there is a continuous extension $\bar{f}:\bar{X}\to\Sigma_{p}$ of $f$.
Then $\phi=p_{0}\bar{f}$ and, hence, $\alpha=\bar{f}^{*}(1)$ where
$1\in\mathbb{Z}[\frac{1}{p}]=\check{H}^{1}(\Sigma_{p})$. Thus, $\alpha$ is
$p$-divisible. ∎

###  2.3. Lipschitz extension

The following fact is well-known. It can be proven by use of Helly’s theorem
and the transfinite induction [CGM],[Dr].

######  2.4 Proposition.

For any metric space $X$ and any $\lambda$-Lipschitz map $f:A\to Y$, $A\subset
X$, where $Y=\mathbb{R}$ or $Y=[a,b]$ there is a $\lambda$-Lipschitz extension
$\bar{f}:X\to Y$.

##  3\. Main Theorem

######  3.1 Theorem.

Every simply connected proper geodesic metric space $X$ admits an embedding of
its Higson compactification into the product of adelic solenoids

$F:\bar{X}\to\prod_{\mathcal{A}}\Sigma_{\infty}$

that induces an isomorphism of 1-dimensional Čech cohomology.

We denote as $[X,Y]_{0}$ pointed homotopy classes of maps
$f:(X,x_{0})\to(Y,y_{0})$.

######  3.2 Proposition.

Let $x_{0}\in X$ be a base point of a simply connected metric space $X$ such
that its inclusion $x_{0}\to\bar{X}$ is a cofibration and let
$0\in\Sigma_{p}$, $p\in\mathbb{N}\cup\\{\infty\\}$, and $1\in S^{1}$ be the
corresponding base points. Then

$[\bar{X},\Sigma_{p}]_{0}=[\bar{X},S^{1}]_{0}=\check{H}^{1}(\bar{X}).$

###### Proof.

The map $p_{0}:Z\to S^{1}$ defines the map
$\Psi:[\bar{X},\Sigma_{p}]_{0}\to[\bar{X},S^{1}]_{0}$ as $\Psi([f])=[fp_{0}]$
for the pointed homotopy class $[f]$ of a map
$f:(\bar{X},x_{0})\to(\Sigma_{p},0)$. We define the inverse map $\Phi$ as
follows. Given a map $\phi:(\bar{X},x_{0})\to(S^{1},1)$ we consider its lift
$\bar{f}:(\bar{X},x_{0})\to(\Sigma_{p},0)$ with respect to $p_{0}$ constructed
in the proof of Proposition 2.3. If $\phi$ has a pointed homotopy to a map
$\phi^{\prime}$, then the lifting of that homotopy will be a pointed homotopy,
since the fiber $p_{0}^{-1}(1)$ is disconnected. This proves $\Phi$ is well-
defined. Clearly, $\Psi\Phi=1$. The uniqueness of lift of connected space $X$
with the prescribed value $0$ for the base point implies that $f$ is the only
lift of $fp_{0}$. Thus, $\Phi\Psi=1$.

Since $x_{0}\to X$ is a cofibration, we obtain
$[\bar{X},S^{1}]_{0}=[\bar{X},S^{1}]=\check{H}^{1}(\bar{X})$. ∎

We can choose a metrics on the solenoids $\Sigma_{p}$,
$p\in\mathbb{N}\cup\\{\infty\\}$, in such a way that $p_{0}:\Sigma_{p}\to
S^{1}$ is 1-Lipschitz and the inclusion map $\mathbb{R}\to\Sigma_{p}$ is
Lipschitz.

For a slowly oscillating map $f:X\to\Sigma_{p}$ we denote by
$\bar{f}:\bar{X}\to\Sigma_{p}$ its extension to the Higson corona. Thus,
$\overline{g|_{X}}=g$ for any continuous map $g:\bar{X}\to\Sigma_{p}$. We note
that for connected space $X$ with $x_{0}\in X$ any continuous map
$f:(X,x_{0})\to(\Sigma_{p},0)$ lands in the subgroup
$\mathbb{R}\subset\Sigma_{p}$.

######  3.3 Proposition.

Let $X$ be a simply connected proper metric space with a base point. For any
elements $[f],[g]\in[\bar{X},\Sigma_{p}]_{0}$, where $\bar{X}$ is the Higson
compactification, and for any $n\in\mathbb{N}$ we have

$n\Psi([f])=\Psi([\overline{nf|_{X}}])$ and

$\Psi([f])+\Psi([g])=\Psi([\overline{f|_{X}+g|_{X}}])$.

###### Proof.

We recall that the restriction of $p_{0}:\Sigma_{p}\to S^{1}$ to $\mathbb{R}$
equals the exponential map $\exp(t)=e^{2\pi it}$. Then

$n\Psi([f])=n[p_{0}f]=[(p_{0}f)^{n}]=[\overline{(p_{0}f|_{X})^{n}}]=[\overline{(exp(f|_{X}))^{n}}]=$
$[\overline{exp(nf|_{X})}]=[\overline{p_{0}(nf|_{X})}]=[p_{0}\overline{nf|_{X}}]=\Psi([\overline{nf|_{X}}])$

and

$\Psi([f])+\Psi([g])=[p_{0}f]+[p_{0}g]=[\overline{p_{0}f|_{X}}]+[\overline{p_{0}f|_{X}}]=[\overline{p_{0}f|_{X}}\cdot\overline{p_{0}g|_{X}}]=$
$[\overline{exp(f|_{X})\cdot
exp(g|_{X})}]=[\overline{exp(f|_{X}+g|_{X})}]=[\overline{p_{0}(f|_{X}+g|_{X})}]=\Psi([\overline{f|_{X}+g|_{X}}]).$

Here the multiplication $\cdot$ is taken in the unit circle $S^{1}$. ∎

For subsets $A$ and $B$ of a metric space $(X,d)$ we denote by

$dist(A,B)=\inf\\{d(a,b)\mid a\in A,\ b\in B\\}$

the distance from $A$ to $B$.

######  3.4 Proposition.

Let $X$ be a proper metric space with a base point $x_{0}\in X$, let $c\in\nu
X$ be a fixed point, and let $A,B\subset\nu X$ be two closed disjoint subsets.
For any $z\in A_{p}\setminus\\{0\\}$, $p\in\mathbb{N}\cup\\{\infty\\}$, there
is a slow oscillating function $f_{z}:X\to\mathbb{R}\subset\Sigma_{p}$ with
the continuous extension $\bar{f}_{z}:\bar{X}\to\Sigma_{p}$ such that
$f_{z}(x_{0})=0$, $\bar{f}_{z}(c)=z$, and
$\bar{f}_{z}(A)\cap\bar{f}_{z}(B)=\emptyset$.

###### Proof.

Suppose that $c\notin B$. Then we enlarge $A$ by adding $c$ to it. We will
define slowly oscillating function $f_{z}:X\to\mathbb{R}\subset\Sigma_{p}$
such that $\bar{f}_{z}$ takes $B$ to $0$ and takes $A$ to $z$.

Let $A^{\prime},B^{\prime}\subset X$ be closed disjoint subsets with
$A=Cl_{\bar{X}}(A^{\prime})\cap\nu X$ and $B=Cl_{\bar{X}}(B^{\prime})\cap\nu
X$. Let

$r_{n}=dist(A^{\prime}\setminus B(x_{0},2^{n}),B^{\prime}\setminus
B(x_{0},2^{n})).$

In view of Proposition 2.2 we may assume that $r_{n}$ is increasig with
$r_{n}\to\infty$.

We consider the partition of $X\setminus B(x_{0},1)$ into closed annuli
$A_{n}$ centered at $x_{0}$ with outer radius $2^{n+1}$ and with the inner
radius $2^{n}$. Consider

$C=A^{\prime}\cap\bigcup_{n=1}^{\infty}A_{2n-1}\ \ \text{and}\ \
C^{\prime}=A^{\prime}\cap\bigcup_{n=1}^{\infty}A_{2n}.$

Since $A^{\prime}=C\cup C^{\prime}$ and $c\in Cl_{\bar{X}}(A^{\prime})$,
either $c\in Cl_{\bar{X}}(C)$ or $c\in Cl_{\bar{X}}(C^{\prime})$. Without loss
of generality we may assume the first.

Let $z_{n}\in\mathbb{R}$ be a sequence converging to $z$ in $\Sigma_{p}$. We
may assume that $z_{n}\to\infty$ in $\mathbb{R}$ and
$z_{n}\leq\min\\{\sqrt{r_{n}},2^{n-1}\\}$. We define $f_{z}(C_{n})=z_{n}$ and
$f_{z}(B^{\prime})=0$. Let

$\lambda_{n}=\max\\{\frac{1}{\sqrt{r_{n}}},\frac{1}{2^{n}}\\}.$

Note that the restriction of $f_{z}$ to $(C\cup B^{\prime})\cap A_{2n-1}$ is
$\lambda_{n}$-Lipschitz. Indeed, for $x\in B^{\prime}$ and $y\in C_{n}$ we
have $d(x,y)\geq r_{n}$ and

$|f_{z}(y)-f_{z}(x)|=z_{n}\leq\sqrt{r_{n}}\leq\frac{1}{\sqrt{r_{n}}}d(x,y).$

By Proposition 2.4 the above restriction of $f_{z}$ can be extended to a
$\lambda_{n}$-Lipschitz map

$f_{z}^{2n-1}:A_{2n-1}\to[0,z_{n}].$

We show that the union map

$f^{2n-1}_{z}\cup f^{2n+1}_{z}:A_{2n-1}\cup A_{2n+1}\to[0,z_{n+1}]$

is $\lambda_{n}$-Lipschitz. For if, $x\in A_{2n+1}$ and $y\in A_{2n-1}$ we
have $d(x,y)\geq 2^{2n}$ and, hence,

$|f^{2n+1}_{z}(x)-f^{2n-1}_{z}(y)|\leq z_{n+1}\leq
2^{n}=\frac{1}{2^{n}}2^{2n}\leq\frac{1}{2^{n}}d(x,y)\leq\lambda_{n}d(x,y).$

By Proposition 2.4 the map $f^{2n-1}_{z}\cup f^{2n+1}_{z}:A_{2n-1}\cup
A_{2n+1}\to\mathbb{R}$ can be extended to $\lambda_{n}$-Lipschitz map
$f^{2n}_{z}:A_{2n}\to\mathbb{R}$. We extend $f_{z}^{0}$ continuously to
$B(x_{0},1)$ with $f_{z}(x_{0})=0$. The union $\cup_{k}f_{z}^{k}$ defines a
slow oscillating function $f_{z}:X\to\mathbb{R}$ with the required properties.
Indeed, since the inclusion of the subgroup $\mathbb{R}\to\Sigma_{p}$ is a
Lipschitz function, the map $f_{z}:X\to\Sigma_{p}$ is slowly oscillating. Then
by Proposition 2.1 there is a continuous extension
$\bar{f}_{z}:\bar{X}\to\Sigma_{p}$. Note that $\bar{f}_{z}(B)=0$. Since the
sequence $z_{k}\subset\mathbb{Z}=\mathbb{R}\cap A_{p}\subset\Sigma_{p}$
converges to $z$, we obtain $\bar{f}_{z}(c)=\bar{f}_{z}(A)=z\neq 0$. ∎

A family of elements $\\{x_{\alpha}\\}\subset A$ of an abelian group is called
linearly independent if for any finite subfamily $\\{x_{\alpha_{i}}\\}$
satisfying the equality $\sum_{i}n_{i}x_{\alpha_{i}}=0$ with
$n_{i}\in\mathbb{Z}$ it follows that $n_{i}=0$ for all $i$.

######  3.5 Proposition.

The group $A_{p}/\mathbb{Z}$, $p\in\mathbb{N}\cup\\{\infty\\}$, contains a
family of linearly independent elements $\\{u_{\alpha}\\}$ of cardinality
continuum.

###### Proof.

First we show it for finite $p$. The group of $p$-adic rational numbers
$\mathbb{Q}_{p}$ contains $A_{p}$ and $\mathbb{Q}$ as subgroups with
$A_{p}\cap\mathbb{Q}=\mathbb{Z}$. The group $\mathbb{Q}_{p}$ is divisible and,
hence, is a vector space over $\mathbb{Q}$. Thus,
$\mathbb{Q}_{p}=(\mathbb{Q}_{p}/\mathbb{Q})\oplus\mathbb{Q}$. Let
$\\{v_{\alpha}\\}\subset\mathbb{Q}_{p}$ be a basis for the summand
$\mathbb{Q}_{p}/\mathbb{Q}$. Then for each $v_{\alpha}$ there is $k=k(\alpha)$
such that $p^{k}v_{\alpha}\in A_{p}$. We define
$u_{\alpha}=p^{k(\alpha)}v_{\alpha}$. Suppose that $\sum
n_{i}u_{i}\in\mathbb{Z}$. Then $\sum
n_{i}p^{k(i)}v_{i}\in\mathbb{Z}\subset\mathbb{Q}$. Therefore,
$n_{i}p^{k(i)}=0$ for all $i$. Hence $n_{i}=0$ for all $i$. Thus the family
$\\{u_{\alpha}\\}$ is linearly independent over $\mathbb{Z}$.

We note that $A_{\infty}\cong\prod_{p}A_{p}$ where the product is taken over
all prime numbers $p$. We take a family $\\{z_{\alpha}\\}\subset A_{\infty}$
in such way that $\pi_{p}(z_{\alpha})=u_{\alpha}$ where
$\pi_{p}:\prod_{p}A_{p}\to A_{p}$ is the projection onto the factor. Then
$\\{z_{\alpha}\\}$ forms a linearly independent family in
$A_{\infty}/\mathbb{Z}$. ∎

Proof of Theorem 3.1. We fix points $x_{0}\in X$ and $c\in\nu X$. Since $X$ is
a reasonably nice space, we may assume that the inclusion $x_{0}\to X$ is a
cofibration. Otherwise we attach to $X$ an interval $[x_{0},x_{1}]$ along some
$x_{1}\in X$.

Let $\\{z_{\alpha}\in A_{\infty}\mid\alpha\in{\mathcal{A}}^{\prime}\\}$ be the
family form Proposition 3.5 of the cardinality of continuum $\mathfrak{c}$.

It is not difficult to show that the weight of the Higson corona $\nu X$ is
$\mathfrak{c}$. Let $\mathcal{U}$ be a basis of topology of $\nu X$ of
cardinality $\mathfrak{c}$. Define

$\mathcal{P}=\\{(U,U^{\prime})\in\mathcal{U}\times\mathcal{U}\mid\bar{U}\cap\bar{U}^{\prime}=\emptyset\\}$

and let $\gamma:\mathcal{P}\to\mathcal{A}^{\prime}$ be a bijection. For every
pair $(U,U^{\prime})\in\mathcal{P}$ we apply Proposition 3.4 with $A=\bar{U}$,
$B=\bar{U}^{\prime}$ and $z=z_{\gamma(U,U^{\prime})}$ to obtain a function
$f_{z}:X\to\mathbb{R}$ that extends to a map
$\bar{f}_{z}:(\bar{X},x_{0})\to(\Sigma_{p},0)$ with $f_{z}(x_{0})=0$,
$\bar{f}_{z}(c)=z$, and $\bar{f}_{z}(A)\cap\bar{f}_{z}(B)=\emptyset$.

We denote by $f_{\alpha}=f_{z_{\alpha}}$, $\alpha\in\mathcal{A}^{\prime}$. We
claim that the cohomology classes
$\\{[p_{0}\bar{f}_{\alpha}]\\}_{\alpha\in\mathcal{A}^{\prime}}$ in
$\check{H}^{1}(\bar{X})$ are linearly independent. Suppose that $\sum
n_{i}[p_{0}\bar{f}_{\alpha_{i}}]=0$. Then by Proposition 3.3,

$\sum n_{i}[p_{0}\bar{f}_{\alpha_{i}}]=\sum
n_{i}\Psi[\bar{f}_{\alpha_{i}}]=\sum\Psi[\overline{n_{i}f_{\alpha_{i}}}]=\Psi([\overline{\sum
n_{i}f_{\alpha_{i}}}])=0.$

Hence there is a rel $x_{0}$ homotopy of $p_{0}\overline{\sum
n_{i}f_{\alpha_{i}}}$ to the constant map to $1\in S^{1}$. The lift of this
homotopy is a rel $x_{0}$ homotopy in $\Sigma_{\infty}$ of $\overline{\sum
n_{i}f_{\alpha_{i}}}$ to the constant map to $0$. Therefore, $\sum
n_{i}\bar{f}_{\alpha_{i}}(c)=\sum n_{i}z_{\alpha_{i}}$ lies in the path
component of $0$. Thus, $\sum n_{i}z_{\alpha_{i}}\in\mathbb{Z}$. The linear
independence of $\\{z_{\alpha}+\mathbb{Z}\\}\in A_{\infty}/\mathbb{Z}$ implies
that all $n_{i}=0$.

We complete the cohomology classes
$\\{[p_{0}\bar{f}_{\alpha}]\\}_{\alpha\in\mathcal{A}^{\prime}}$ in
$\check{H}^{1}(\bar{X})=\oplus\mathbb{Q}$ to a basis
$\\{[\phi_{\alpha}]\\}_{\alpha\in\mathcal{A}^{\prime\prime}}$ where
$\phi_{\alpha}:(\bar{X},x_{0})\to(S^{1},1)$. Let
$\bar{f}_{\alpha}:(\bar{X},x_{0})\to(\Sigma_{\infty},0)$ be the lift of
$\phi_{\alpha}$ defined by the map $\Phi$ of Proposition 3.2. Let
$\mathcal{A}=\mathcal{A}^{\prime}\cup\mathcal{A}^{\prime\prime}$. Since the
map

$F=(\bar{f}_{\alpha})_{\alpha\in\mathcal{A}}:\bar{X}\to\prod_{\mathcal{A}}\Sigma_{\infty}$

restricted to the Higson corona $\nu X$ separates the disjoint closures of
basis sets, it separates distinct points and, hence, it is an embedding on
$\nu X$. We note that the restriction
$F^{\prime}=F|_{X}:X\to\prod_{\mathcal{A}}\mathbb{R}\subset\prod_{\mathcal{A}}\Sigma_{\infty}$
to $X$ lands in $\prod_{\mathcal{A}}\mathbb{R}$.

We approximate $F$ by embedding as follows. Fix $\omega$ coordinates
$\alpha_{1},\dots,\alpha_{n},\dots\in\mathcal{A}$ and consider $\|\
\|_{\infty}$ metric on $\mathbb{R}^{\omega}$. We denote by
$f_{i}=f_{\alpha_{i}}$. Let

$d_{i}(t)=\sup_{x\in X\setminus B(x_{0},t)}diam(f_{i}(B(x,1))).$

Since $f_{i}$ is extendible to the Higson corona, it is slowly oscillating.
Hence $\lim_{t\to\infty}d_{i}(t)=0$. We recall that the limitation topology on
a functional space $C(X,Y)$ with metrizable $Y$ is defined by the collection
of open balls $B_{\rho}(f,1)$ with respect to all compatible metrics $\rho$ on
$Y$ where the ball in $C(X,Y)$ is defined by means of the sup metric [Bo]. We
note that the functional space $C(X,\mathbb{R}^{\omega})$ taken with the
limitation topology is a Baire space [T],[Bo]. This allows to prove that the
map $f=(f_{i})_{i=1}^{N}:X\to\mathbb{R}^{\omega}$ can be approximated by
embedding $g:X\to\mathbb{R}^{\omega}$ in such a way that

$\rho(t)=\sup_{x\in X\setminus B(x_{0},t)}\|f(x)-g(x)\|_{\infty}\to 0$

as $t\to\infty$. Then the $i$th coordinate of $g$, the map
$g_{i}:X\to\Sigma_{p}$, is slowly oscillating for each $i$. Indeed, by the
triangle inequality

$\lim_{t\to\infty}\sup_{x\in X\setminus
B(x_{0},t)}diam(g_{i}(B(x,1)))\leq\lim_{t\to\infty}(d_{i}(t)+2\rho(t))=0.$

Note that the extension $\bar{g}_{i}:\bar{X}\to\Sigma_{\infty}$ of $g_{i}$
coincides with $\bar{f}_{i}$ on $\nu X$.

We replace the map $F$ with a new map

$F^{\prime}=(\bar{g}_{i})_{i=1}^{\infty}\times(\bar{f}_{\alpha})_{\alpha\in\mathcal{A}\setminus\\{\alpha_{1},\dots,\alpha_{n}\dots\\}}:\bar{X}\to\prod_{\mathcal{A}}\Sigma_{\infty}$

which is an embedding. ∎

######  3.6 Remark.

All coordinate maps $\bar{f}_{\alpha}$ in the embedding
$F:\bar{X}\to\prod_{\mathcal{A}}\Sigma_{\infty}$ have the property that the
restriction $\bar{f}_{\alpha}$ to $X$ lands in the subgroup
$\mathbb{R}\subset\Sigma_{\infty}$. Moreover, by the construction
$\bar{f}_{\alpha}(X)\subset\mathbb{R}_{+}$.

##  4\. Embedding into the product of Knaster continua

The proof of Theorem 3.1 in the case of finite $p$ brings the following.

######  4.1 Theorem.

For any $p$ every simply connected proper geodesic metric space $X$ admits an
embedding of its Higson compactification into the product of $p$-adic
solenoids

$F:\bar{X}\to\prod_{\mathcal{A}}\Sigma_{p}$

that induces a rational isomorphism of 1-cohomology,

$\check{H}^{1}(\prod_{\mathcal{A}}\Sigma_{p})\otimes\mathbb{Q}\to\check{H}^{1}(\bar{X})\otimes{\mathbb{Q}}=\check{H}^{1}(\bar{X})=\bigoplus_{\mathcal{A}}\mathbb{Q}.$

We call a map $f:X\to Z$ essential if every map $g:X\to Z$ homotopic to $f$ is
surjective. We call a subset $X\subset\prod Z_{\alpha}$ essential if its
projection on each factor $p_{\alpha}:X\to Z_{\alpha}$ is essential. Since the
embedding $F:\bar{X}\to\prod\Sigma_{p}$ in Theorem 4.1 is essential, we obtain

######  4.2 Corollary.

For any $p$ every simply connected proper geodesic metric space $X$ admits an
essential embedding of its Higson compactification into a product of $p$-adic
solenoids.

######  4.3 Proposition.

For any $p\in\\{\infty\\}\cup\mathbb{N}\setminus\\{1\\}$ any surjective map
$f:Y\to\Sigma_{p}$ of a connected compact Hausdorff space is essential.

###### Proof.

Assume that $f$ is homotopic to $g$ which is not onto. Then for some
projection $\pi:\Sigma_{p}\to S^{1}$ onto the circle in the inverse limit
presentation of $\Sigma_{p}$ the composition $\pi\circ g$ is not onto. Hence
it is homotopic to a constant map. The lifting of this homotopy defines a
homotopy of $g$ to the disconnected fiber $A_{p}$ and hence to a point. Thus,
we have a homtopy of $f$ to a constant map. Since $\Sigma_{p}$ has more than
one path component, this is impossible for a surjecive map. ∎

We define a continuum $K_{p}=\Sigma_{p}/\sim$ to be the quotient space under
the identification $x\sim-x$. A theorem of Bellamy [Be] says that $K_{2}$ is
homeomorphic to the Knaster continuum, also known as the Bucket handle
continuum. We note that the subgroup $\mathbb{R}\subset\Sigma_{p}$ is taken
under this identification to a path component of $K_{p}$ which is an injective
image of the ray $\mathbb{R}_{+}$ with the initial point $q(0)$ where
$q:\Sigma_{p}\to K_{p}$ is the quotient map.

Since for complex numbers the conjugation commutes with taking the power,
$\bar{z}^{n}=\overline{z^{n}}$, we have a commutative diagram

$\begin{CD}S^{1}@<{z^{n}}<{}<S^{1}\\\ @V{q}V{}V@V{q}V{}V\\\
I@<{\bar{n}}<{}<I\\\ \end{CD}$

where $q$ is the projection onto the interval, the orbit space of the
conjugation, and $\bar{n}$ is the induced map. Note that $\bar{n}:I\to I$ is
the $n$ times folding the interval. Thus, there is the inverse limit
presentation of $K_{p}$ that fits into the diagram:

$\setcounter{MaxMatrixCols}{11}\begin{CD}S^{1}@<{z^{n_{1}}}<{}<S^{1}@<{z^{n_{2}}}<{}<S^{1}@<{z^{n_{3}}}<{}<\dots
@<{}<{}<\Sigma_{p}\\\ @V{q}V{}V@V{q}V{}V@V{q}V{}V@V{q}V{}V\\\
I@<{\bar{n}_{1}}<{}<I@<{\bar{n}_{2}}<{}<I@<{\bar{n}_{3}}<{}<\dots
@<{}<{}<K_{p}.\\\ \end{CD}$

We identify the first interval in the bottom inverse sequence with $[0,1]$,
the second with $[0,n_{1}]$, the third with $[0,n_{1}n_{2}]$ and so on to view
the bonding maps as ’isometric’ folding. Moreover, we view each folding map
$\bar{n}_{k}:[0,n_{1}n_{2}\cdots n_{k}]$ as a retraction onto
$[0,n_{1}n_{2}\cdots n_{k-1}]$. Thus

$\mathbb{R}_{+}=\bigcup_{k}[0,n_{1}n_{2}\cdots n_{k}]$

is naturally embedded into $K_{p}$ as the path component of $q(0)$.

######  4.4 Proposition.

For any $p\in\\{\infty\\}\cup\mathbb{N}\setminus\\{1\\}$ any surjective map
$f:Y\to K_{p}$ of a compact Hausdorff space is essential.

###### Proof.

Let $q:\Sigma_{p}\to K_{p}$ be the quotient map. Note that $q$ is open and all
preimages $q^{-1}(y)$ are 2-point sets except $q^{-1}(q(0))=0$. We show that
the compact space $Z$ in the pull-back in the diagram

$\begin{CD}Z@>{f^{\prime}}>{}>\Sigma_{p}\\\ @V{q^{\prime}}V{}V@V{q}V{}V\\\
Y@>{f}>{}>K_{p}\\\ \end{CD}$

is connected. Suppose that $Z=U_{1}\cup U_{2}$ is a disjoint union of nonempty
open sets. For $i=1,2$ we define subsets $A_{i}=\\{y\in
Y\mid(q^{\prime})^{-1}(y)\subset U_{i}\\}$. Since $U_{i}$ are closed and open
and $q^{\prime}$ isan open map the sets $A_{1}=Y\setminus q^{\prime}(U_{2})$
and $A_{2}=Y\setminus q^{\prime}(U_{1})$ are closed and open. Since
$q^{\prime}$ has single point fibers at least one of $A_{i}$ is not empty, let
it be $A_{1}$. Since $Y$ is connected, we obtain $Y=A_{1}$. Then
$U_{2}=\emptyset$ which contradicts to the assumption.

Assume that $f$ is homotopic by means of homotopy $H:Y\times I\to K_{p}$ to a
non-surjective map $g$. Note that for $y\in Y$ with $f(y)\in\mathbb{R}_{+}$ we
have $H(y\times I)\subset\mathbb{R}_{+}$. We may assume that the inner metric
on $\mathbb{R}_{+}$ induced by a metric on $K_{p}$ is the standard metric on
$\mathbb{R}_{+}$. Thus $\mathbb{R}_{+}$ makes infinitely many waves in $K_{p}$
of amplitude 1. Then the compactness of $Y$ and the continuity of $H$ imply
that there is an upper bound $b$ on the diameter of a path
$H(y,-):I\to\mathbb{R}_{+}$ in $\mathbb{R}_{+}$. Let $Y_{0}=f^{-1}([0,2b])$
and let $a\in K_{p}\setminus g(Y)$. We consider a new homotopy $H_{0}:Y\times
I\to K_{p}$ defined by the region under the graph of a function
$\phi:Y\to[0,1]$ with $\phi(Y_{0})=0$ and $\phi(f^{-1}(a))=1$. Thus, the
homotopy $H_{0}$ is stationary on $Y_{0}$ and does not move other points
through $0\in\mathbb{R}_{+}\subset K_{p}$. The homotopy $H_{0}$ defines a
homotopy $H_{0}^{\prime}:Z\times I\to K_{p}$. Let
$Z_{0}=(q^{\prime})^{-1}(Y_{0})$. By the homotopy lifting property of a
fibration $q|:\Sigma_{p}\setminus\\{0\\}\to K_{p}\setminus\\{0\\}$ there is a
lift $\bar{H}_{0}:(Z\setminus Z_{0})\times I\to\Sigma_{p}$ of
$H^{\prime}_{0}|_{(Y\setminus Y_{0})\times I}$. Since $q$ is open, the
extension of $\bar{H}_{0}$ to $\bar{H}_{1}:Z\times I\to\Sigma_{p}$ by the
stationary homotopy of $f^{\prime}|_{Z_{0}}$ is a homotopy of
$f^{\prime}:Z\to\Sigma_{p}$ to a map with the image in a proper subset. This
contradicts to Proposition 4.3. ∎

######  4.5 Theorem.

For any $p$ and any simply connected finite dimensional proper geodesic metric
space $X$ its Higson compactification can be essentially embedded into the
product of continua $K_{p}$.

###### Proof.

The embedding follows from the fact that the family of functions
$\mathcal{C}=\\{q\bar{f}_{\alpha}\mid\alpha\in\mathcal{A}\\}$, where
$q:\Sigma_{p}\to K_{p}$ is the quotient map, separates disjoint closures of
basis sets by the argument of Proposition 3.4.

In view of Proposition 4.4 all maps $q\bar{f}_{\alpha}$ are essential. ∎

## References

  * [Be] D. Bellamy, A tree-like continuum without the fixed point property, Houston Math. J. 6 (1979), 1-13.
  * [Bo] Ph. Bowers, Limitation Topologies on Function Spaces, Trans. Amer Math. Soc. V. 314, No 1 (1989), 421-431.
  * [BV] J. Burgos, A. Verjovsky, Adelic solenoid I: Structure and topology, ArXiv:1603.05676v4 [math.CV] 2 Aug 2019.
  * [CGM] A. Connes, M. Gromov, H. Moscovici, Group cohomology with Lipschitz control and higher signatures, GAFA v. 3 (1993), 1-78.
  * [Dr] A. Dranishnikov, Asymptotic topology, Russian Math. Surveys 55 (2000), no. 6, 1085-1129.
  * [DF] A. Dranishnikov, S. Ferry, On the Higson-Roe corona, Russian Math. Surveys 52 (1997), no. 5, 1017-1028.
  * [DFW] A. Dranishnikov, S. Ferry, and S. Weinberger, An Etale Approach to the Novikov Conjecture Comm.Pure Appl. Math, 61 no.2, 2008, 139-155.
  * [DKU] A. Dranishnikov, J. Keesling, V. Uspenski, On the Higson corona of uniformly contractible spaces, Topology 37 (1998), no. 4, 791-–803.
  * [HR] Higson, Nigel; Roe, John Analytic $K$-homology. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford, 2000.
  * [K] J. Keesling, The one-dimensional Čech cohomology of the Higson compactification and its corona. Topology Proc. 19 (1994), 129-–148.
  * [Ro1] Roe, John, Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer. Math. Soc. 104 (1993), no. 497.
  * [Ro2] Roe, John, Lectures on coarse geometry. University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003.
  * [T] H. Torunczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247-262.
  * [Yu] Guoliang Yu, _The Novikov conjecture for groups with finite asymptotic dimension_ , Ann. of Math. (2) 147 (1998), no. 2, 325–355.