# Multiplicities in the length spectrum and growth rate of Salem numbers

Alexandr Grebennikov  A. Grebennikov: IMPA, Estrada Dona Castorina 110, Jardim
Botanico, Rio de Janeiro, Brazil<EMAIL_ADDRESS>
###### Abstract.

We prove that multiplicities in the length spectrum of a non-compact
arithmetic hyperbolic orbifold of dimension $n\geqslant 4$ have exponential
growth rate

$\langle g(L)\rangle\geqslant c\frac{e^{([n/2]-1)L}}{L^{1+\delta_{5,7}(n)}},$

extending the analogous result for even dimensions of Belolipetsky, Lalín,
Murillo and Thompson. Our proof is based on the study of (square-rootable)
Salem numbers.

As a counterpart, we also prove a refined version of these asymptotic results,
adapting the argument of Götze and Gusakova for the case of square-rootable
Salem numbers, to show that one can not obtain a better estimate on the
multiplicities using our approach.

##  1\. Introduction

The length spectrum of a Riemannian manifold is defined as the list of lengths
of its closed geodesics written in ascending order and with multiplicities.
The length spectrum has received a lot of attention because it encodes a lot
of properties of the manifold and is closely related to the eigenvalue
spectrum of its Laplace-Beltrami operator.

The multiplicities in the length spectrum are believed to reflect how rigid
the manifold is. A generic Riemannian manifold is not supposed to have any
additional structure, thus any non-trivial multiplicities [1]. Non-arithmetic
hyperbolic manifolds represent intermediate situation: their multiplicities
are known to be unbounded in dimensions $2$ ([17]) and $3$ ([15]), and mean
multiplicities can sometimes grow exponentially [3], though with a small
constant in the exponent.

Current note is devoted to “the structured” case: arithmetic hyperbolic
orbifolds, which may be defined as quotients of hyperbolic space
$\operatorname{\mathbb{H}}^{n}$ by an action of an arithmetic lattice
$\Gamma\subset\operatorname{Isom}(\operatorname{\mathbb{H}}^{n})$. Our Theorem
1.1 states that mean multiplicities in the length spectrum of a non-compact
arithmetic hyperbolic orbifold of dimension $n\geqslant 4$ have exponential
growth rate. Analogous statements in dimensions $2$ ([4], [5]) and $3$ ([14])
are classical, in these cases even sharper bounds are known and non-
compactness is not required. The case of even dimension at least $4$ is due to
Belolipetsky, Lalín, Murillo and Thompson [2, Proposition 3], we adapt their
approach to work in odd dimensions as well and present a uniform proof for
both cases for the sake of transparency.

######  Theorem 1.1.

Let $\mathcal{O}$ be a non-compact arithmetic hyperbolic orbifold of dimension
$n\geqslant 4$, $G(L)$ — the set of its closed geodesics of length at most
$L$. Then for sufficiently large $L$

$\langle g(L)\rangle=\frac{|G(L)|}{|\\{\ell(\gamma):\gamma\in
G(L)\\}|}\geqslant c\frac{e^{([n/2]-1)L}}{L^{1+\delta_{5,7}(n)}},$

where $c=c(n)>0$ is a constant, and $\delta_{5,7}(n)=\begin{cases}1,&\text{ if
}n=5,7,\\\ 0,&\text{otherwise.}\end{cases}$

As usual in the arithmetic case, our proof makes use of several number-
theoretic concepts. A Salem number is a real algebraic integer $\lambda>1$,
such that it is conjugate to $\lambda^{-1}$ and its remaining conjugates lie
on the unit circle. Note that we allow Salem numbers to have degree $2$, to be
consistent with [2] and [6].

Next we need to define a more technical notion.

######  Definition 1.2 (see Section 1.3 of [6]).

A Salem number $\lambda$ is called square-rootable over a field
$K\subset\operatorname{\mathbb{Q}}(\lambda+\lambda^{-1})$ if there exist a
totally positive $\alpha\in K$ and a monic palindromic polynomial $q$, such
that even degree coefficients of $q$ lie in $K$, odd degree coefficients of
$q$ lie in $\sqrt{\alpha}K$, and

$q(x)q(-x)=p(x^{2}),$

where $p$ is the minimal polynomial of $\lambda$ over $K$.

The following result of Emery, Ratcliffe and Tschantz relates (square-
rootable) Salem numbers to the length spectrum of arithmetic hyperbolic
orbifolds.

######  Theorem 1.3 (Theorems 1.1 and 1.6 of [6]).

Let $\Gamma\subset\operatorname{Isom}(\operatorname{\mathbb{H}}^{n})$ be an
arithmetic lattice of the simplest type defined over a totally real number
field $K$, $\gamma\in\Gamma$ — a hyperbolic element, $\ell(\gamma)$ — the
length of the geodesic in $\operatorname{\mathbb{H}}^{n}/\Gamma$ corresponding
to it. If $n$ is even then $\lambda=e^{\ell(\gamma)}$ is a Salem number, with
$\deg_{K}(\lambda)\leqslant n+1$. If $n$ is odd then $e^{2\ell(\gamma)}$ is a
Salem number, square-rootable over $K$, with $\deg_{K}(\lambda)\leqslant n+1$.

Since in our setting the orbifold
$\mathcal{O}=\operatorname{\mathbb{H}}^{n}/\Gamma$ is non-compact, it is thus
of the simplest type defined over $K=\operatorname{\mathbb{Q}}$ (see [12,
Sections 1-2]).

As Theorem 1.3 suggests, estimates on the order of growth of (square-rootable)
Salem numbers are the crucial ingredient in our proof of Theorem 1.1,
contained in Propositions 3.1 and 3.2. These propositions are quite elementary
and not especially precise, but they are sufficient for the main purposes on
our work.

However, sometimes it may be interesting to refine these bounds and to obtain
a sharp asymptotic formula for the number of (square-rootable) Salem numbers
of given degree on the initial segment of the real line. For the case of all
Salem numbers (which corresponds to orbifolds of even dimension) this was
shown by Götze and Gusakova in [9, Theorem 1.1], restated below as Theorem
4.1. In Section 4 we modify their approach for the case of square-rootable
Salem numbers and prove the following Theorem 1.4.

######  Theorem 1.4.

Denote by $\operatorname{Sal}_{m}^{sq}(R)$ the set of square-rootable (over
$\operatorname{\mathbb{Q}}$) Salem numbers of degree $2m$ up to $R$. Then for
fixed $m\geqslant 3$ and $R\to\infty$

  1. (1)

if $m=3,4$, then

$|\operatorname{Sal}_{m}^{sq}(R)|=w_{m-1}\frac{6}{\pi^{2}}R^{m/2}\log(R)\cdot\operatorname{err}(R),$

  2. (2)

if $m\geqslant 5$, then

$|\operatorname{Sal}_{m}^{sq}(R)|=w_{m-1}\frac{\zeta(\frac{1}{2}[\frac{m+1}{2}])}{\zeta([\frac{m+1}{2}])}R^{m/2}\cdot\operatorname{err}(R),$

where the error term satisfies

$\begin{cases}\operatorname{err}(R)=1+o(1),&\text{ when $m$ is odd; }\\\
\varepsilon+o(1)\leqslant\operatorname{err}(R)\leqslant 1+o(1),&\text{ when
$m$ is even. }\end{cases}$

Here $w_{m}$ are explicit constants defined by formula (4),
$\varepsilon=\varepsilon(m)>0$ is an implicit constant, and $\zeta$ is the
Riemann zeta function.

This theorem extends asymptotic formula for $m=2$ proved in [2, Theorem 2]

$|\operatorname{Sal}_{2}^{sq}(R)|=\frac{4}{3}R^{3/2}+O(R)$

to larger degrees.

The main motivation for Theorem 1.4 is to ensure that Proposition 3.2 is
essentially tight, and therefore one can not obtain a better bound in Theorem
1.1 with our approach. Also this theorem may be used to write down an explicit
expression for the constant $c(n)$ in Theorem 1.1, see Remark 3.4.

Acknowledgements. The author is grateful to his advisor Mikhail Belolipetsky
for bringing his attention to this problem and helpful comments that have
significantly improved this note.

##  2\. Notation and preliminaries

###  2.1. Salem numbers

A polynomial $p$ is called palindromic if it satisfies $p(x)=x^{\deg
p}p(x^{-1})$. It is not hard to prove that the minimal polynomial of a Salem
number has to be palindromic of even degree. See [8] for a general survey on
Salem numbers.

Notation $\operatorname{\mathbb{A}}$ is used for the ring of algebraic
integers over $\operatorname{\mathbb{Q}}$. We denote by
$\operatorname{Sal}_{m}\subset\operatorname{\mathbb{A}}$ the set of Salem
numbers of degree $2m$, and let

$\operatorname{Sal}_{m}(Q)=\operatorname{Sal}_{m}\;\cap\;(1,Q].$

Similarly, we denote by
$\operatorname{Sal}^{sq}_{m}\subset\operatorname{\mathbb{A}}$ the set of
square-rootable (over $\operatorname{\mathbb{Q}}$) Salem numbers of degree
$2m$, and let

$\operatorname{Sal}_{m}^{sq}(Q)=\operatorname{Sal}_{m}^{sq}\;\cap\;(1,Q].$

###  2.2. Asymptotic behavior of functions

For real-valued functions $f(x)$ and $g(x)$ we use the following standard
notation to compare their asymptotic behavior as $x\to\infty$.

  * •

$f(x)=O(g(x))$ if $|f(x)|\leqslant Cg(x)$;

  * •

$f(x)=o(g(x))$ if for any $\varepsilon>0$ there exists $N=N(\varepsilon)$ such
that $|f(x)|\leqslant\varepsilon g(x)$ for $x\geqslant N$;

  * •

$f(x)\sim g(x)$ if $f(x)-g(x)=o(g(x))$.

Here we emphasize that implicit constants $C$ and $N=N(\varepsilon)$ are
allowed to depend additionally on the dimension of the manifold (usually
denoted by $n$) or degree of the Salem number (usually denoted by $2m$), but
not on anything else.

For $x\in\operatorname{\mathbb{R}}$ we use $\log(x)$ for the natural (base
$e$) logarithm of $x$, and $[x]$ for the largest integer not exceeding $x$.

###  2.3. Arithmetic orbifolds of the simplest type

To give the definitions, we follow the exposition of [6, Section 2.3]. A
quadratic form $f$ in $n+1$ variables over a totally real number field $K$ is
called admissible if it has signature $(n,1)$, and for each non-trivial
embedding $\sigma:K\to\operatorname{\mathbb{R}}$ corresponding quadratic form
$f^{\sigma}$ over $\sigma(K)$ is positive definite. For a subring
$R\subset\operatorname{\mathbb{R}}$ denote by $\operatorname{O}(f,R)$ the
group of matrices preserving this quadratic form

$\operatorname{O}(f,R)=\\{T\in\operatorname{GL}(n+1,R)\mid f(Tx)=f(x)\text{
for all }x\in\operatorname{\mathbb{R}}^{n+1}\\},$

and let $\operatorname{O}^{\prime}(f,R)$ be its subgroup, also keeping both
connected components of the negative cone
$\\{x\in\operatorname{\mathbb{R}}^{n+1}\mid f(x)<0\\}$ invariant.

Let $f_{n}$ be the standard Lorentzian form in $n+1$ variables

$f_{n}(x)=x_{1}^{2}+\ldots+x_{n}^{2}-x_{n+1}^{2}.$

A subgroup $\Gamma\subset\operatorname{Isom}(\operatorname{\mathbb{H}}^{n})$
is called an arithmetic subgroup of the simplest type defined over $K$ if
there exist $M\in\operatorname{GL}(n+1,\operatorname{\mathbb{R}})$ and an
admissible quadratic form $f$ over $K$ in $n+1$ variables such that

  * •

$f(Mx)=f_{n}(x)$ for any $x\in\operatorname{\mathbb{R}}^{n+1}$;

  * •

subgroups $M\Gamma M^{-1}$ and $\operatorname{O}^{\prime}(f,\mathfrak{o}_{K})$
of $\operatorname{O}^{\prime}(f,\operatorname{\mathbb{R}})$, where
$\mathfrak{o}_{K}$ is the ring of integers of $K$, are commensurable (that is,
their intersection has finite index in both of them).

Such $\Gamma$ is always a discrete subgroup of
$\operatorname{Isom}(\operatorname{\mathbb{H}}^{n})$ of finite covolume. The
hyperbolic orbifold $\operatorname{\mathbb{H}}^{n}/\Gamma$ in this case is
also called arithmetic of the simplest type defined over $K$.

As mentioned in the introduction, for the case of non-compact arithmetic
hyperbolic orbifolds this construction is actually exhaustive.

######  Proposition 2.1 ([12], Sections 1-2).

Any non-compact arithmetic hyperbolic orbifold is of the simplest type defined
over $\operatorname{\mathbb{Q}}$.

This allows to carry out our analysis only for the case
$K=\operatorname{\mathbb{Q}}$. In particular, for the remaining part of the
note square-rootable Salem numbers are assumed to be square-rootable over
$\operatorname{\mathbb{Q}}$ in the sense of Definition 1.2.

##  3\. Proof of Theorem 1.1

######  Proposition 3.1.

For a fixed $m\in\operatorname{\mathbb{N}}$ and $Q\to\infty$,

$\operatorname{Sal}_{m}(Q)=O(Q^{m}).$

###### Proof.

It suffices to estimate the number of monic palindromic polynomials
$p\in\operatorname{\mathbb{Z}}[x]$ of degree $2m$, such that all its roots
except one have absolute value at most $1$, and the remaining one has absolute
value at most $Q$. Suppose

$p(x)=x^{2m}+p_{2m-1}x^{2m-1}+\ldots+p_{1}x+1,\;p_{k}=p_{2m-k}$

is such a polynomial. Rewriting it via its roots as

$p(x)=(x-\beta_{1})(x-\beta_{2})\ldots(x-\beta_{2m}),\;\;\;|\beta_{1}|\leqslant
Q,\;|\beta_{i}|\leqslant 1\text{ for }2\leqslant i\leqslant 2m,$

we obtain that

$|p_{2m-k}|\leqslant\sum_{1\leqslant i_{1}<\ldots<i_{k}\leqslant
2m}|\beta_{i_{1}}\ldots\beta_{i_{k}}|\leqslant{2m\choose k}\cdot Q=O(Q).$

So, we have $O(Q)$ options for each of the coefficients $p_{1},\ldots,p_{m}$ —
therefore, $O(Q^{m})$ such polynomials. ∎

######  Proposition 3.2.

For a fixed $m\in\operatorname{\mathbb{N}}$ and $Q\to\infty$

$\operatorname{Sal}_{m}^{sq}(R)=O(R^{m/2}\cdot\eta(R,m)),\text{ where }$
$\eta(R,m)=\begin{cases}R^{1/2},&\text{ if }m=1,2;\\\ \log(R),&\text{ if
}m=3,4;\\\ 1,&\text{ if }m\geqslant 5.\end{cases}$ (1)

###### Proof.

If $\lambda$ is a square-rootable Salem number with the minimal polynomial
$p$, then roots of $p(x^{2})$ are $\pm\lambda^{1/2},\;\pm\lambda^{-1/2}$ (all
with multiplicity one) and the other roots of absolute value $1$. Let $q$ be
the polynomial from the definition of square-rootable Salem number 1.2. Then
all roots of $q$ except one have absolute value at most $1$, and the remaining
one has absolute value at most $Q=R^{1/2}$. Suppose

$q(x)=q^{2m}+q_{2m-1}x^{2m-1}+\ldots+q_{1}x+1,\;q_{k}=q_{2m-k},$

then we can estimate its coefficients as

$q_{0}=q_{2n}=1,\;|q_{k}|\leqslant{2m\choose k}\cdot Q\leqslant cQ.$ (2)

On the other hand, all the coefficients of $q$ are algebraic integers because
they can be polynomially expressed via roots of $p(x^{2})$.

Multiplication of $\alpha$ by a square in $\operatorname{\mathbb{Q}}$ does not
change the set $\sqrt{\alpha}\operatorname{\mathbb{Q}}$, therefore we may
assume $\alpha$ to be a square-free positive integer. So, even degree
coefficients of $q$ lie in
$\operatorname{\mathbb{Q}}\cap\operatorname{\mathbb{A}}=\operatorname{\mathbb{Z}}$,
and odd degree ones lie in

$\sqrt{\alpha}\operatorname{\mathbb{Q}}\cap\operatorname{\mathbb{A}}=\\{\sqrt{\alpha
r^{2}}:r\in\operatorname{\mathbb{Q}},\;\sqrt{\alpha
r^{2}}\in\operatorname{\mathbb{A}}\\}=\\{\sqrt{\alpha
r^{2}}:r\in\operatorname{\mathbb{Q}},\alpha
r^{2}\in\operatorname{\mathbb{A}}\\}=\sqrt{\alpha}\operatorname{\mathbb{Z}}.$

In particular, by (2) there is no reason to consider $\sqrt{\alpha}>cQ$, since
it would force all the odd degree coefficients to be zero.

Hence there are at most $O(Q)$ possible options for even degree coefficients
of $q$ and (if we fix a square-free integer $\alpha\in[1,(cQ)^{2}]$)
$O(Q/\sqrt{\alpha})$ options for odd degree ones. Summing it up, the total
number of possible polynomials $q(x)$ is at most

$C\cdot
Q^{[m/2]}\cdot\sum^{(cQ)^{2}}_{\alpha=1}\big{(}\frac{Q}{\sqrt{\alpha}}\big{)}^{[(m+1)/2]}=C\cdot
Q^{[m/2]+[(m+1)/2]}\cdot\sum^{(cQ)^{2}}_{\alpha=1}\frac{1}{\sqrt{\alpha}^{[\frac{m+1}{2}]}}=O(Q^{m}\cdot\eta(Q^{2},m)).$

Recalling that $Q=R^{1/2}$ concludes the proof. ∎

###### Proof of Theorem 1.1.

By the result of Gangolli and Warner [7, Proposition 5.4], generalizing the
classical theorem of Margulis [13] to the setting of non-compact orbifolds, we
can estimate the number of closed geodesics of length up to $L$ as

$|G(L)|\sim\frac{e^{(n-1)L}}{(n-1)L}.$ (3)

Case of even $n$: Theorem 1.3 asserts that the exponents $e^{\ell(\gamma)}$ of
geodesic lengths are exactly Salem numbers of degree at most $n+1$ (over
$\operatorname{\mathbb{Q}}$, by Proposition 2.1). In fact, the degree is at
most $n$ since degrees of Salem numbers are always even. By Proposition 3.1 we
have $|\operatorname{Sal}_{m}(e^{L})|=O(e^{mL})$, and therefore,

$|\\{\ell(\gamma):\gamma\in
G(L)\\}|\leqslant\sum^{n/2}_{m=1}|\operatorname{Sal}_{m}(e^{L})|=O(e^{nL/2}).$

Combining with formula (3) gives

$\langle g(L)\rangle\geqslant c(n)\frac{e^{(n/2-1)L}}{L}.$

Case of odd $n$: Again, by Theorem 1.3 and Proposition 2.1 the exponent
$e^{2\ell(\gamma)}$ is a square-rootable over $\operatorname{\mathbb{Q}}$
Salem number of degree at most $n+1$. By Proposition 3.2 we have

$|\operatorname{Sal}_{m}^{sq}(e^{2L})|=O(e^{mL}\cdot\eta(e^{2L},m)),$

and therefore,

$|\\{\ell(\gamma):\gamma\in
G(L)\\}|\leqslant\sum_{m\leqslant(n+1)/2}|\operatorname{Sal}^{sq}_{m}(e^{2L})|=O\big{(}e^{(n+1)L/2}\cdot\eta(e^{2L},(n+1)/2)\big{)}.$

Since in this case $(n+1)/2\geqslant 3$, the definition of $\eta$ yields

$\eta(e^{2L},(n+1)/2)=O(L^{\delta_{5,7}(n)}),$

and combining it with formula (3) gives

$\langle g(L)\rangle\geqslant
c(n)\frac{e^{((n-1)/2-1)L}}{L^{1+\delta_{5,7}(n)}}.$

∎

######  Remark 3.3.

The argument we used to prove Theorem 1.1 is quite straightforward. First we
use the Prime Geodesic Theorem to count the number of closed geodesics in
$\operatorname{\mathbb{H}}^{n}/\Gamma$ up to a given length. Then we recall
that length of a geodesic in this situation can be expressed as an exponent of
a (square-rootable) Salem number. Finally, we estimate the number of (square-
rootable) Salem numbers of given degree on the initial segment of the real
line.

The last part here is not perfectly linked to the first two, what gives some
room for improvement: though it is true that any (square-rootable) Salem
number arises from length of a geodesic in
$\operatorname{\mathbb{H}}^{n}/\Gamma$ for some arithmetic lattice $\Gamma$ of
the simplest type (see [6, Theorems 1.1 and 1.6]), it seems unlikely that all
the (square-rootable) Salem numbers can be generated by a single lattice.

Indeed, analogous bounds for mean multiplicities in the case $n=3$, obtained
by Marklof in [14] by more careful analysis, are stronger than what is
possible to get using our approach. Unfortunately, his methods are very
specific to $3$-dimensional case.

######  Remark 3.4.

The proof of Theorem 1.1 we presented does not intend to optimize the value of
the constant factor $c(n)$ arising from Proposition 3.1 for even dimensions
and Proposition 3.2 for odd dimensions. However, it is possible (and
straightforward) to write down an explicit expression for it if one uses
result of Götze and Gusakova 4.1 for even dimensions and our Theorem 1.4 for
odd dimensions instead. More precisely, for fixed $n\geqslant 4$ and
$L\to\infty$

$\langle
g(L)\rangle\geqslant\big{(}c^{\prime}(n)+o(1)\big{)}\frac{e^{([n/2]-1)L}}{L^{1+\delta_{5,7}(n)}},$
$\text{ where
}\;\;c^{\prime}(n)=\frac{1}{w_{[\frac{n+1}{2}]-1}}\cdot\begin{cases}1,&\text{
if $n$ is even;}\\\ \frac{\pi^{2}}{6},&\text{ if $n=5,7$;}\\\
\frac{\zeta([\frac{n+3}{4}])}{\zeta(\frac{1}{2}[\frac{n+3}{4}])},&\text{ if
$n\geqslant 9$ is odd.}\end{cases},$

and $w_{m}$ are defined by the formula (4). Moreover, one can show that

$w_{m}\leqslant\frac{4(2\sqrt{\pi})^{m-1}}{\sqrt{(m+1)!}},$

and therefore $c^{\prime}(n)$ tends to infinity superexponentially with
respect to $n$.

##  4\. Growth rate of square-rootable Salem numbers

In this section we adapt the argument of Götze and Gusakova [9] to prove
Theorem 1.4 and conclude that the bound from Proposition 3.2 is essentially
tight.

Before heading to the proof, we outline the strategy used by Götze and
Gusakova in [9] to obtain the asymptotic formula 4.1 for the number of Salem
numbers up to $Q$ (all of them, not only square-rootable ones).

######  Theorem 4.1 (Theorem 1.1 of [9]).

For fixed $m$ and $Q\to\infty$

$|\operatorname{Sal}_{m}(Q)|=w_{m-1}Q^{m}+O(Q^{m-1}),$

where

$w_{m}=\frac{2^{m(m+1)}}{m+1}\prod_{k=0}^{m-1}\frac{(k!)^{2}}{(2k+1)!}.$ (4)

Let $V^{all}_{m}$ be the set of monic palindromic polynomials in
$\operatorname{\mathbb{R}}[x]$ of degree $2m$, identified with a subset of
$\operatorname{\mathbb{R}}^{m}$ via considering their coefficients at
$x,x^{2},\ldots,x^{m}$. Define an injective map

$f:\\{(\lambda,\theta_{1},\ldots,\theta_{m-1})\in\operatorname{\mathbb{R}}^{m}\mid
1<\lambda\leqslant
Q,\;0\leqslant\theta_{1}\leqslant\ldots,\leqslant\theta_{m-1}\leqslant\pi\\}\to
V^{all}_{m}\;\text{ as }$
$(\lambda,\theta_{1},\ldots,\theta_{m-1})\mapsto(x-\lambda)(x-\lambda^{-1})\prod_{j=1}^{m-1}(x-e^{i\theta_{j}})(x-e^{-i\theta_{j}}).$
(5)

Then each integer point inside
$V_{m}=\operatorname{Im}f\subset\operatorname{\mathbb{R}}^{m}$ corresponds to
some integer polynomial, with one of its roots being a Salem number. Since the
number of reducible integer polynomials inside $V_{m}$ is $O(Q^{m-1})$ (Lemma
2.2 of [9]), almost all integer points are exactly minimal polynomials of
Salem numbers.

To have a chance to count the number of integer points inside, one needs to
check that the boundary of $V_{m}$ is “nice enough”.

######  Definition 4.2.

We say that the boundary $\partial B$ of a set
$B\subset\operatorname{\mathbb{R}}^{m}$ is of Lipschitz class $(M,L)$ if there
exist $M$ maps

$\varphi_{1},\ldots,\varphi_{M}:[0,1]^{m-1}\to\operatorname{\mathbb{R}}^{m},$

such that each of them is $L$-Lipschitz

$\lVert\varphi_{i}(x)-\varphi_{i}(y)\rVert\leqslant L\lVert x-y\rVert\text{
for any }x,y\in[0,1]^{m-1},\;1\leqslant i\leqslant M$

and $\partial B$ is covered by the union of their images.

######  Lemma 4.3 (Lemma 2.1 of [9]).

Boundary of $V_{m}$ is of Lipschitz class $(M,cQ)$ for some constants $M,c>0$.

Then lattice point counting result of Widmer allows to estimate the number of
lattice points inside $V_{m}$ by its volume:

######  Theorem 4.4 (Widmer [19]).

Let $\Lambda$ be a lattice in $\operatorname{\mathbb{R}}^{d}$ with successive
minima $\lambda_{1},\ldots,\lambda_{d}$,
$B\subset\operatorname{\mathbb{R}}^{d}$ — a bounded set with boundary of
Lipschitz class $(M,L)$. Then $B$ is measurable, and

$\big{|}\\#(\lambda\cap
B)-\frac{\operatorname{Vol}(B)}{\det\Lambda}\big{|}\leqslant
c(d)M\max_{0\leqslant i<d}\frac{L^{i}}{\lambda_{1}\ldots\lambda_{i}}.$

So, the problem is reduced to computation of the volume
$v_{m}=\operatorname{Vol}(V_{m})$. Careful calculation of Jacobian of the map
$f$ (Lemma 2.4 of [9]) and its integration over the domain provide the
estimate

$v_{m}=w_{m-1}Q^{m}+O(Q^{m-1}),$ (6)

thus finishing the proof of Theorem 4.1.

Now return to our Theorem 1.4. We want to count the number of square-rootable
(over $\operatorname{\mathbb{Q}}$) Salem numbers of degree $2m$ up to $Q$,
each of them has an associated polynomial $q(x)$ from Definition 1.2. The
following Definition 4.5 specifies which exactly polynomials we are interested
in.

######  Definition 4.5.

Denote by $P_{m}$ the set of pairs $(\alpha,q)$, where $\alpha$ is a positive
square-free integer and $q$ is a polynomial in $\operatorname{\mathbb{R}}[x]$
such that

  1. (1)

$q$ is monic and palindromic;

  2. (2)

$\deg q=2m$;

  3. (3)

even degree coefficients of $q$ lie in $\operatorname{\mathbb{Z}}$, odd degree
coefficients of $q$ lie in $\sqrt{\alpha}\operatorname{\mathbb{Z}}$;

  4. (4)

all the roots of $q$ except two have absolute value exactly $1$, and the
remaining two are positive reals $\lambda>1$ and $\lambda^{-1}<1$ (both with
multiplicity one);

  5. (5)

$q(\xi)\neq 0$ for any root of unity $\xi$ of degree at most $4m$.

Also define a map $\Phi:P_{m}\to\operatorname{\mathbb{R}}$ by

$\Phi:(\alpha,q)\mapsto\lambda^{2},\text{ where }\lambda\text{ is the unique
root of $q$ with absolute value greater than $1$}.$

We start to examine the set $P_{m}$ by demonstrating that polynomial $q$
determines the value of $\alpha$ uniquely.

######  Lemma 4.6.

If $(\alpha,q)\in P_{m}$ then at least one odd degree coefficient of $q$ is
non-zero. As a consequence, if $(\alpha_{1},q),\;(\alpha_{2},q)\in P_{m}$,
then $\alpha_{1}=\alpha_{2}$.

###### Proof.

Suppose that all odd degree coefficient of $q$ are zero, then $q(x)=q(-x)$. In
particular, if $\lambda$ is a root of $q$, then so is $-\lambda$,
contradiction with condition (4).

For the second part, if $\alpha_{1},\alpha_{2}$ are distinct positive square-
free integers, then all the odd degree coefficients of $q$ have to lie in
$\sqrt{\alpha_{1}}\operatorname{\mathbb{Z}}\cap\sqrt{\alpha_{2}}\operatorname{\mathbb{Z}}=\\{0\\}$,
contradiction with the first part of this lemma. ∎

The next two lemmas assert that $\Phi$ is almost a bijection onto the set of
square-rootable (over $\operatorname{\mathbb{Q}}$) Salem numbers of degree
$2m$.

######  Lemma 4.7.

The image of $\Phi$ coincides with $\operatorname{Sal}_{m}^{sq}$.

######  Lemma 4.8.

For any $x\in\operatorname{\mathbb{R}}$

$|\Phi^{-1}(x)|\leqslant C,$

where $C=C(m)$ is a constant depending only on $m$. Moreover, when $m$ is odd
we may take $C(m)=1$. In other words, if $m$ is odd then $\Phi$ is injective.

###### Proof of Lemma 4.7.

First we check that the image of $\Phi$ lies inside
$\operatorname{Sal}_{m}^{sq}$. Given $(\alpha,q)\in P_{m}$, conditions (1)-(3)
ensure that

$q(x)q(-x)=p(x^{2})$

for some monic palindromic polynomial $p\in\operatorname{\mathbb{Z}}[x],\;\deg
p=2m$. Let $\lambda$ be the root of $q$ from condition (4) (of Definition
4.5), then the roots of $p$ are $\lambda^{2},\lambda^{-2}$ (both with
multiplicity one) and other ones of absolute value $1$.

So, $\lambda^{2}=\Phi((\alpha,q))$ is indeed a Salem number of degree at most
$2m$. To see that it is exactly $2m$, we need to check that $p$ is the minimal
polynomial of $\lambda^{2}$. Suppose it is reducible: $p(x)=p_{1}(x)p_{2}(x)$
for some $p_{1},p_{2}\in\operatorname{\mathbb{Z}}[x]$. Then one of
$p_{1},p_{2}$ has roots only of absolute value $1$, and the classical result
of Kronecker [11] forces it to be cyclotomic. Therefore $p(\xi_{1})=0$ for
some root of unity $\xi_{1}$ of degree at most $2m$, thus $q(\xi_{2})=0$ for
some root of unity $\xi_{2}$ of degree at most $4m$, what contradicts
condition (5).

Now we check that the $\operatorname{Sal}^{sq}_{m}$ is contained in the image
of $\Phi$. Suppose $\lambda$ is a square-rootable Salem number with minimal
polynomial $p$, $\deg p=2m$. Consider $\alpha\in\operatorname{\mathbb{Q}}$ and
$q\in\operatorname{\mathbb{R}}[x]$ from Definition 1.2. Multiplication of
$\alpha$ by a square in $\operatorname{\mathbb{Q}}$ does not change the set
$\sqrt{\alpha}\operatorname{\mathbb{Q}}$, therefore we may assume that
$\alpha$ is indeed a square-free positive integer. The polynomial $q$
satisfies conditions (1) and (2) automatically. Since

$q(x)q(-x)=p(x^{2}),$

the roots of $q(x)q(-x)$ are $\pm\lambda^{1/2},\pm\lambda^{-1/2}$ (all with
multiplicity one) and other ones of absolute value $1$. Recall that a
palindromic polynomial has root $z$ if and only if it has root $z^{-1}$, what
allows us to conclude (by replacing $q(x)$ with $q(-x)$ if necessary) that $q$
satisfies condition (4) as well.

Condition (5) follows from irreducibility of $p$, and to check condition (3)
we repeat the argument used in the proof of Proposition 3.2: coefficients of
$q$ can be polynomially expressed via its roots — so, they are algebraic
integers; thus, even degree coefficients of $q$ lie in
$\operatorname{\mathbb{Q}}\cap\operatorname{\mathbb{A}}=\operatorname{\mathbb{Z}}$,
odd degree coefficients of $q$ lie in

$\sqrt{\alpha}\operatorname{\mathbb{Q}}\cap\operatorname{\mathbb{A}}=\\{\sqrt{\alpha
r^{2}}:r\in\operatorname{\mathbb{Q}},\;\sqrt{\alpha
r^{2}}\in\operatorname{\mathbb{A}}\\}=\\{\sqrt{\alpha
r^{2}}:r\in\operatorname{\mathbb{Q}},\alpha
r^{2}\in\operatorname{\mathbb{A}}\\}=\sqrt{\alpha}\operatorname{\mathbb{Z}}.$

So, we constructed $(\alpha,q)\in P_{m}$, such that
$\Phi((\alpha,q))=\lambda$. ∎

###### Proof of Lemma 4.8.

If $\Phi((\alpha,q))$ is a Salem number with minimal polynomial $p$, then
$q(x)q(-x)=p(x^{2})$, thus the size of preimage is bounded above by the number
of such decompositions of $p(x^{2})$. But such decomposition is determined by
choosing one representative from each pair of opposite roots, therefore we may
take $C(m)=2^{2m}$.

To prove injectivity of $\Phi$ in the case of odd $m$ more careful analysis is
required. Fix a Salem number $\lambda^{2}$ of degree $2m$ in the image of
$\Phi$. Suppose that $\Phi((\alpha,q))=\lambda^{2}$ for some $(\alpha,q)\in
P_{m},\;\alpha>1$, then $\lambda$ is the root of $q$ with maximal absolute
value. Condition (3) allows to write

$q(x)=q_{2m}x^{2m}+\sqrt{\alpha}q_{2m-1}x^{2m-1}+q_{2m-2}x^{2m-2}+\ldots,\;\text{
with }q_{i}\in\operatorname{\mathbb{Q}}.$

Substituting $\lambda$ and separating even and odd terms gives

$\sqrt{\alpha}=\frac{-q_{2m}\lambda^{2m}-q_{2m-2}\lambda^{2m-2}-\ldots}{q_{2m-1}\lambda^{2m-1}+q_{2m-3}\lambda^{2m-3}+\ldots}\in\operatorname{\mathbb{Q}}(\lambda).$

Note that denominator of this fraction is non-zero: indeed,
$\deg_{\operatorname{\mathbb{Q}}}\lambda\geqslant\deg_{\operatorname{\mathbb{Q}}}\lambda^{2}=2m$,
and we are not allowed to have all the odd degree coefficients equal to zero
by Lemma 4.6.

Case 1. $\Phi((1,q_{1}))=\lambda^{2}$ for some $(1,q_{1})\in P_{m}$.

Then $q_{1}\in\operatorname{\mathbb{Q}}[x]$ is the minimal polynomial of
$\lambda$ over $\operatorname{\mathbb{Q}}$, thus there is no other pair with
the first coordinate equal to $1$ in $\Phi^{-1}(\lambda^{2})$. Suppose that
$\Phi((\alpha,q))=\lambda^{2}$ for some $(\alpha,q)\in P_{m},\;\alpha>1$. From
$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}]=\deg q_{1}=2m$
and
$\operatorname{\mathbb{Q}}(\sqrt{\alpha})\subset\operatorname{\mathbb{Q}}(\lambda)$
we deduce that

$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}(\sqrt{\alpha})]=m,\;\;\text{which
is an odd number.}$

Let $r_{1}$ be the minimal polynomial of $\lambda$ over
$\operatorname{\mathbb{Q}}(\sqrt{\alpha})$, $\deg r_{1}=m$. Then
$r_{-1}(x)=x^{m}r_{1}(x^{-1})$ is the minimal polynomial of $\lambda^{-1}$
over $\operatorname{\mathbb{Q}}(\sqrt{\alpha})$, and $\deg r_{-1}=m$ as well.
Since they are minimal, and
$q_{1},q\in\operatorname{\mathbb{Q}}(\sqrt{\alpha})[x]$ have $\lambda$ and
$\lambda^{-1}$ as their roots by condition (4), we have

$r_{1},r_{-1}\mid q_{1},\;\;r_{1},r_{-1}\mid q.$

Conditions (4) and (5) imply that $q$ has only two real roots. Since the
degrees of $r_{1}$ and $r_{-1}$ are odd, $\lambda$ is the unique real root of
$r_{1}$ and $\lambda^{-1}$ is the unique real root of $r_{-1}$. In particular,
$r_{1}\neq r_{-1}$. Since they are both irreducible over
$\operatorname{\mathbb{Q}}(\sqrt{\alpha})$, this implies that they are
coprime, and

$r_{1}r_{-1}\mid q_{1},\;r_{1}r_{-1}\mid q.$

From $\deg r_{1}r_{-1}=\deg q=\deg q_{1}=2m$ we have $q=q_{1}$. But
$\alpha\neq 1$, contradiction with Lemma 4.6.

Case 2. $\Phi((1,q_{1}))\neq\lambda^{2}$ for any $(1,q_{1})\in P_{m}$.
Equivalently,
$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}]\neq 2m$.

Since $\lambda^{2}$ is a Salem number of degree $2m$, the degree of
$\operatorname{\mathbb{Q}}(\lambda)$ over $\operatorname{\mathbb{Q}}$

$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}]=[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}(\lambda^{2})]\cdot[\operatorname{\mathbb{Q}}(\lambda^{2}):\operatorname{\mathbb{Q}}]$

is either $4m$ or $2m$. Since the latter option is ruled out by the
assumption, only the first one remains. Suppose that

$\Phi((\alpha,q))=\Phi((\alpha^{\prime},q^{\prime}))=\lambda^{2}\;\text{ for
some }(\alpha,q),(\alpha^{\prime},q^{\prime})\in
P_{m},\;\alpha,\alpha^{\prime}>1.$

This implies, as noted above, that
$\sqrt{\alpha},\sqrt{\alpha^{\prime}}\in\operatorname{\mathbb{Q}}(\lambda)$,
and

$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}(\sqrt{\alpha})]=[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}(\sqrt{\alpha^{\prime}})]=2m.$

Then $q\in\operatorname{\mathbb{Q}}(\sqrt{\alpha})[x]$ and
$q^{\prime}\in\operatorname{\mathbb{Q}}(\sqrt{\alpha^{\prime}})[x]$ are the
minimal polynomials of $\lambda$ over
$\operatorname{\mathbb{Q}}(\sqrt{\alpha})$ and, respectively,
$\operatorname{\mathbb{Q}}(\sqrt{\alpha^{\prime}})$. Hence
$\alpha=\alpha^{\prime}$ would imply $q=q^{\prime}$, and we may assume
$\alpha\neq\alpha^{\prime}$.

The remaining part of the argument is similar to the previous case. The degree

$[\operatorname{\mathbb{Q}}(\lambda):\operatorname{\mathbb{Q}}(\sqrt{\alpha},\sqrt{\alpha^{\prime}})]=m$

is odd, thus the minimal polynomials $r_{1}$ and $r_{-1}$ of $\lambda$ and,
respectively, $\lambda^{-1}$ over
$\operatorname{\mathbb{Q}}(\sqrt{\alpha},\sqrt{\alpha^{\prime}})$ do not
coincide, and therefore are coprime. By their minimality (and condition (4)
for $q$ and $q^{\prime}$) both $r_{1}$ and $r_{-1}$ divide $q$ and
$q^{\prime}$, hence so does their product $r_{1}r_{-1}$. From

$\deg r_{1}r_{-1}=\deg q=\deg q^{\prime}=2m$

we now deduce that $q=q^{\prime}$. But $\alpha\neq\alpha^{\prime}$,
contradiction with Lemma 4.6. ∎

######  Remark 4.9.

This sort of ambiguity for even $m$ in Lemma 4.8 is the reason why we were not
able to obtain the exact value of multiplicative constant in Theorem 1.4 for
even $m$. Note that $\Phi$ actually can be non-injective: for example, the
root of Salem polynomial
$p(x)=x^{8}-56x^{7}-157x^{6}-228x^{5}-247x^{4}-228x^{3}-157x^{2}-56x+1$ has
four preimages in $P_{4}$ with the values of $\alpha$ equal to $2,6,26$ and
$78$, respectively. It seems unlikely for such phenomenon to occur “too
often”, but we were unable to make this statement rigorous.

The following lemma will help us to handle condition (5) while counting
polynomials in the set $P_{m}$ from Definition 4.5.

######  Lemma 4.10.

Suppose that the boundary $\partial B$ of some bounded set
$B\subset\operatorname{\mathbb{R}}^{m}$ is of Lipschitz class $(M,L)$, and
$H\subset\operatorname{\mathbb{R}}^{m}$ can be covered by a finite family of
hyperplanes $H_{1},\ldots,H_{K}$. Then the boundary of

$B^{\prime}=B\setminus H$

is of Lipschitz class $(MK+M,L)$.

###### Proof.

Consider the maps $\\{\varphi_{i}\\}_{1\leqslant i\leqslant M}$ covering
$\partial B$ by their images. Define $\\{\varphi_{ij}\\}_{1\leqslant
i\leqslant M,1\leqslant j\leqslant K}$ as the composition of $\varphi_{i}$
with orthogonal projection onto $H_{j}$. Since the projections do not increase
distances, these maps are also $L$-Lipschitz. If $x\in\partial
B^{\prime}\setminus\partial B$, then $x$ lies in the interior of $B$
intersected with one of the hyperplanes $H_{j_{0}}$. Since $B$ is bounded, the
orthogonal line to $H_{j_{0}}$ passing through $x$ has to intersect the
boundary $\partial B$. Therefore $\partial B^{\prime}$ is covered by the
images of $\\{\varphi_{i}\\}$ and $\\{\varphi_{ij}\\}$. ∎

Also we record two simple facts about sums of powers of square-free integers
(see [18], [10], [16, OEIS/A005117]).

######  Proposition 4.11.

For $x\to\infty$,

$\sum_{n\in\operatorname{\mathbb{N}},\;n\leqslant x,\;n\text{ is square-
free}}\frac{1}{n}\sim\frac{6}{\pi^{2}}\log(x).$

######  Proposition 4.12.

For $s>1$,

$\sum_{n\in\operatorname{\mathbb{N}},\;n\text{ is square-
free}}n^{-s}=\frac{\zeta(s)}{\zeta(2s)}.$

###### Proof of Theorem 1.4.

Denote $R^{1/2}$ by $Q$. For a positive square-free integer $\alpha$ define

$P_{m,\alpha}=\\{q\in\operatorname{\mathbb{R}}[x]:(\alpha,q)\in
P_{m},\;\Phi((\alpha,q))\leqslant R\\},$

where $P_{m}$ and $\Phi$ are as in Definition 4.5. These sets are disjoint by
Lemma 4.6, and by Lemma 4.7 and Lemma 4.8 we have

$|\operatorname{Sal}_{m}^{sq}(R)|=\sum_{\alpha\in\operatorname{\mathbb{N}},\;\text{square-
free}}|P_{m,\alpha}|\;\;\;\text{ for odd $m$, and }$ (7)
$\frac{1}{C(m)}\big{(}\sum_{\alpha\in\operatorname{\mathbb{N}},\;\text{square-
free}}|P_{m,\alpha}|\big{)}\leqslant|\operatorname{Sal}_{m}^{sq}(R)|\leqslant\sum_{\alpha\in\operatorname{\mathbb{N}},\;\text{square-
free}}|P_{m,\alpha}|\;\;\;\text{ for even $m$.}$ (8)

Notice that these sums are actually finite. Indeed, similarly to the argument
from the proof of Proposition 3.2, we say that $q\in P_{m,\alpha}$ has exactly
one root with absolute value greater than $1$, and its absolute value is at
most $R^{1/2}=Q$. Therefore, absolute values of coefficients of $q$ have to be
bounded above by $c_{0}Q$ for some constant $c_{0}=c_{0}(m)$, and we can not
have $\alpha>(c_{0}Q)^{2}$.

So, the problem is reduced to calculation of the sizes of sets $P_{m,\alpha}$.

Following the strategy of Götze and Gusakova, let $V^{all}_{m}$ be the set of
monic palindromic polynomials in $\operatorname{\mathbb{R}}[x]$ of degree
$2m$, identified with a subset of $\operatorname{\mathbb{R}}^{m}$ via
considering their coefficients at $x,x^{2},\ldots,x^{m}$; and consider the map
$f$ defined by formula (5). Condition (3) suggests us to look at the lattice
$\Lambda_{\alpha}=\sqrt{\alpha}\operatorname{\mathbb{Z}}e_{1}+\operatorname{\mathbb{Z}}e_{2}+\sqrt{\alpha}\operatorname{\mathbb{Z}}e_{3}+\ldots\subset\operatorname{\mathbb{R}}^{m}$.

Again, let $V_{m}\subset V_{m}^{all}$ be the image of $f$. Then the set of
lattice points $V_{m}\cap\Lambda_{\alpha}$ almost coincides with
$P_{m,\alpha}$. To make them exactly the same, we need to satisfy condition
(5), thus exclude from $V_{m}$ the following sets:

$Z(\xi)=\\{q\in V^{all}_{m}\mid q(\xi)=0\\},\text{ where $\xi$ is a root of
unity of degree at most $4m$}.$

Both $\operatorname{Re}(q(\xi))=0$ and $\operatorname{Im}(q(\xi))=0$ are
linear equations on the coefficients of $q$, and at least one of them is non-
trivial — therefore, each $Z(\xi)$ lies inside a hyperplane. Recall that
boundary of $V_{m}$ is of Lipschitz class $(M,cQ)$ by Lemma 4.3, then Lemma
4.10 allows to conclude that the boundary of

$V^{\prime}_{m}=V_{m}\setminus\big{(}\bigcup_{\xi^{d}=1,\;d\leqslant
4m}Z(\xi)\big{)}$

is of Lipschitz class $(M^{\prime},cQ)$ for some constant
$M^{\prime}\leqslant(4m)^{2}M$. Lattice points counting Theorem 4.4 then
yields

$\big{|}\\#P_{m,\alpha}-\frac{\operatorname{Vol}(V^{\prime}_{m})}{\sqrt{\alpha}^{[\frac{m+1}{2}]}}\big{|}=\big{|}\\#(V^{\prime}_{m}\cap\Lambda_{\alpha})-\frac{\operatorname{Vol}(V^{\prime}_{m})}{\sqrt{\alpha}^{[\frac{m+1}{2}]}}\big{|}=O\big{(}\frac{Q^{m-1}}{(\sqrt{\alpha})^{[\frac{m+1}{2}]-1}}\big{)}\;\text{
for }\alpha\leqslant(c_{0}Q)^{2}.$

Recall the volume estimate $\eqref{eq:v_m_w_m}$

$\operatorname{Vol}(V^{\prime}_{m})=\operatorname{Vol}(V_{m})=w_{m-1}Q^{m}+O(Q^{m-1}),$

then

$|P_{m,\alpha}|=w_{m-1}\frac{Q^{m}}{\sqrt{\alpha}^{[\frac{m+1}{2}]}}+O\big{(}\frac{Q^{m-1}}{(\sqrt{\alpha})^{[\frac{m+1}{2}]-1}}\big{)}.$

Taking the sum over square-free positive integers up to $(c_{0}Q)^{2}$ gives

$\sum_{\begin{subarray}{c}1\leqslant\alpha\leqslant(c_{0}Q)^{2},\\\
\alpha\text{ is square-
free}\end{subarray}}|P_{m,\alpha}|=w_{m-1}Q^{m}\cdot\big{(}\sum_{\begin{subarray}{c}1\leqslant\alpha\leqslant(c_{0}Q)^{2},\\\
\alpha\text{ is square-
free}\end{subarray}}\frac{1}{\sqrt{\alpha}^{[\frac{m+1}{2}]}}\big{)}+O\big{(}Q^{m-1}\cdot\sum_{\begin{subarray}{c}1\leqslant\alpha\leqslant(c_{0}Q)^{2},\\\
\alpha\text{ is square-
free}\end{subarray}}\frac{1}{\sqrt{\alpha}^{[\frac{m+1}{2}]-1}}\big{)}$ (9)

Next we use Proposition 4.11 to compute the first term of (9) for $m=3,4$ (and
use a trivial bound $\sum_{n\leqslant x}1/\sqrt{n}=O(\sqrt{x})$ for the second
term):

$\sum_{\begin{subarray}{c}1\leqslant\alpha\leqslant(c_{0}Q)^{2},\\\
\alpha\text{ is square-
free}\end{subarray}}|P_{m,\alpha}|=w_{m-1}Q^{m}\cdot\frac{6}{\pi^{2}}\cdot
2\log(Q)\cdot(1+o(1))+O(Q^{m}).$ (10)

Similarly, we use Proposition 4.12 to compute the first term of (9) for
$m\geqslant 5$:

$\sum_{\begin{subarray}{c}1\leqslant\alpha\leqslant(c_{0}Q)^{2},\\\
\alpha\text{ is square-
free}\end{subarray}}|P_{m,\alpha}|=w_{m-1}Q^{m}\cdot\frac{\zeta(\frac{1}{2}[\frac{m+1}{2}])}{\zeta([\frac{m+1}{2}])}\cdot(1+o(1))+O(Q^{m-1}\log
Q).$ (11)

It remains to recall that $Q=R^{1/2}$ and combine equations (10) and (11) with
equations (7) and (8) to obtain the desired bound.

∎

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# A Stronger Baseline for Automatic Pfirrmann Grading of lumbar spine MRI
using Deep Learning

###### Abstract

This paper addresses the challenge of grading visual features in lumbar spine
MRI using Deep Learning. Such a method is essential for the automatic
quantification of structural changes in the spine, which is valuable for
understanding low back pain. Multiple recent studies investigated different
architecture designs, and the most recent success has been attributed to the
use of transformer architectures. In this work, we argue that with a well-
tuned three-stage pipeline comprising semantic segmentation, localization, and
classification, convolutional networks outperform the state-of-the-art
approaches. We conducted an ablation study of the existing methods in a
population cohort, and report performance generalization across various
subgroups. Our code is publicly available to advance research on disc
degeneration and low back pain.

Index Terms—  Lumbar Spine, Pfirrmann Grading, Disc Degeneration, Deep
Learning, Convolutional Neural Network

##  1 Introduction

Lumbar Disc Degeneration (LDD) was found to be one of several reasons causing
Low Back Pain [1]. Although there are several grading systems to quantify LDD,
the Pfirrmann and Schneiderman scales are commonly used [2]. Pfirrmann et al.
proposed a $5$-scale grading system using T2-weighted MR imaging [3]. The
grading system uses an algorithmic approach for LDD using signal intensity and
disc homogeneity, demonstrated in Fig. 1. Over the past few years, several
Deep Learning based methods were built to quantify LDD using MR imaging and
the Pfirrmann grading system.

(a) PG 2

(b) PG 3

(c) PG 4

(d) PG 5

Fig. 1: Pfirrmann grades in L5-S1 as graded in the Northern Finland Birth
Cohort dataset

In a study by da Silva Barreiro et al.  [4], the intervertebral discs (IVDs)
were extracted using manually segmented binary masks, and an artificial neural
network was used to perform Pfirrmann grading (PG). Later, SpineNet [5] – a
framework that can automatically grade spine MRIs for different spine
disorders, including LDD using the PG system was proposed. The authors used a
graph-based approach to detect and group the vertebral bodies (VB) and discs
in their work. Then, a VGG-M based architecture was used as the classifier.
Newer architectures and methods were subsequently exploited in other works.
For instance, SpineNetV2 [6] used a UNet [7] based architecture to improve
vertebrae detection, and a ResNet [8] architecture was used for the
classification of the spine disorders. Although the improvement was not
substantial, this work highlighted the impact of design choices on the overall
outcome.

More recently, a transformer [9] based architecture for spine grading [10] was
proposed and it outperformed the architecture designs proposed in the earlier
works. In another related work, DeepSpine [11] used a ResNeXt50 [12]
architecture with the UNet model for spine segmentation to detect stenosis.
Additionally, it is noteworthy to see an external validation of these methods
[13].

Fig. 2: An end-to-end pipeline for quantifying LDD in the lumbar spine using
Pfirrmann grading

Although there is continuous improvement to create better spine grading models
with newer architectures, we find a lack of thorough empirical evaluation in
the existing research. Furthermore, current studies leveraged clinical data,
but in reality, population cohorts are used in research. In this study, we
take a step towards addressing these two challenges. Specifically, the
contributions of our work are:

  1. 1.

We revisit the design choices used to create architectures for lumbar spine
MRI grading, and propose a strong convolutional neural network (CNN) based
method that outperforms the existing approaches in a systematic evaluation.

  2. 2.

Compare to the existing methods, we evaluate our methodology on a population
cohort, which reflects the transferability of our results to downstream
research applications.

  3. 3.

Finally, we make the code of this paper and the pre-trained models available
to benefit further research.

##  2 Materials and Methods

###  2.1 The Northern Finland Birth Cohort 1966

The Northern Finland Birth Cohort (NFBC $1966$) dataset contains lumbar spine
MRI data for $1500$ participants born in $1966$. Imaging data were acquired
when the participants turned $46-47$ years old using $1.5$ Tesla GE Signa
HDxt. Each slice has a resolution of $512\times 512$ with a slice thickness of
$4mm$ and a pixel spacing of $0.5mm$.

For $1315$ subjects, the per-disc PGs were provided by a consensus of three
experienced readers. Compared to the original definition, in the NFBC dataset,
PGs 1 and 2 were treated as grade 2 [14]. In addition to the MR imaging, the
participants from the same sub-group of NFBC were followed up with
questionnaires to record information related to LBP.

###  2.2 Spine segmentation dataset

A useful initial step for spine grading (Fig. 2), is to perform localization
of functional spine units (FSUs) [10] – i.e. regions of interest (ROIs),
containing two vertebral bodies and the associated disc. We approached this
problem using semantic segmentation with $12$ classes: $6$ VBs, $5$ discs, and
the background.

An experienced clinician (T.M.), annotated a set of $250$ mid and para-
sagittal slices with detailed boundaries for VBs and the discs ($512\times
512$ pixels; original image resolution). Subsequently, we used an active
learning approach by training UNet++ [15] with EfficientNetB3 [16] to produce
an initial segmentation model.

To extend the available data for model training, we used our initial model to
produce predictions for unlabeled data, re-annotated poorly segmented samples,
and included them in the training set. This process has been repeated several
times to obtain segmentations for the whole dataset.

###  2.3 Lumbar Spine segmentation

An evaluation of segmentation architectures such as FPN [17], UNet [7], and
UNet++ [15] was done through a series of ablation experiments that included
encoder selection and hyperparameter tuning.

We trained our models for $100$ epochs with an initial learning rate of
$0.0001$ and a batch size of $10$ having an input shape of $512\times 512$.
The augmentation pipeline included random brightness, contrast, gamma, $\pm
15$ degrees of rotation, and Gaussian blur. A weight decay of $0.0001$ was
incorporated with the Adam optimizer [18], and Jaccard loss [19] was used for
the optimization.

We split the whole dataset into $2$ portions: $80\%$ for training and
validation, and $20\%$ for independent testing. We then took $20\%$ of the
former set for model selection. The mean Intersection over Union (mIoU) over
independent runs based on $5$ random seeds was used to evaluate the model
performance.

###  2.4 Functional spine unit localization

After finalizing the model architecture for segmentation, a post-processing
step was incorporated to delineate $5$ FSUs from each slice of the lumbar
spine MRI. As such, we defined the FSU ROIs adaptively based on the bounding
boxes inferred from the resulting segmentation mask. Subsequently, we rescaled
these ROIs to $224\times 224$ pixels. Due to the volumetric nature of MRI, we
obtained multiple slices with a shape of $224\times 224$ associated with each
FSU. For efficiency, we utilized a subset of $n$ slices for PG.

(a) Segmentation architectures (b) Classification architectures

Fig. 3: A comparison of different architecture combinations on the
segmentation and classification tasks. (b) $n\in\\{1,4,6,8\\}$ represents the
number of slices.

###  2.5 Pfirrmann grading

We utilized a common CNN to assess the LDD severity of the FSUs extracted in
the previous step. As such, we built a DL architecture consisting of a feature
extraction module for learning the representations of FSU slices, and a multi-
layer perception (MLP) for PG. The candidates for the feature extractor were
variants of ResNet [8] and EfficientNet [16]. The MLP was a parametric non-
linear function mapping the FSU representations to $4$ PGs.

We optimized the focal loss [20], which is defined in the multi-class setting
as

$\mathcal{L}_{\mathrm{Focal}}=-\alpha_{t}(1-p_{t})^{\gamma}\log p_{t},$ (1)

where $p_{t}$ is the predicted probability of class $t$, and $\alpha$ and
$\gamma$ are the loss hyperparameters. In our experiments, we used inverse
class frequency to specify $\alpha$ and set $\gamma=2$.

The data here were also split to have $20\%$ for testing. Similar to
segmentation, we used $20\%$ of the training set for model selection. We
trained all models using the Adam optimizer with a learning rate of $0.001$, a
batch size of $20$, and a weight decay of $0.0001$.

##  3 Experiments

###  3.1 Implementation details

####  3.1.1 Setup

Through a series of ablation studies, we aimed to find the best models for
each stage of our pipeline comprising vertebral segmentation, FSU
localization, and spine grading. The pipeline was decoupled, and the training
was performed independently for each stage. We implemented all the methods
using the PyTorch framework, and conducted all the experiments on $2\ \times$
NVIDIA RTX A4000.

####  3.1.2 Spine grading: Ablation study

To analyze the importance of different components, we conducted ablation
studies over loss functions, samplers, schedulers, and slice volumes. To
process the slice volumes and inputs, we replaced the first layer in the
grading network with a randomly initialized one so that the number of input
channels is equal to $n$.

The base model was further trained with incremental settings such as consensus
labels, ImageNet weights, augmentations, and Test Time Augmentations (TTA). At
each incremental setting, the improvement is recorded. For augmentations, we
used grid distortion & downscaling in addition to the augmentations used for
segmentation. We used a validation set to assess improvements.

###  3.2 Results

####  3.2.1 Lumbar Spine segmentation

A comparison of different architectures is shown in Fig. 3. From model
comparisons, we found similar performance exhibited by UNet++ and FPN with the
ResNet34 encoder. However, UNet++ suffered from high variance, making the FPN
with ResNet34 the model of choice. FPN + ResNet34 combination scored an mIoU
of $97.5\pm 0.1$ over $5$ random seeds.

####  3.2.2 Spine grading model

The base model for spine grading was chosen from the ablation study as
discussed in Section 3.1.2. The results of this can be seen in Fig. 3.
Although EfficientNetB0 architecture with $6$ slices had the best Balanced
Accuracy (BA), the standard error is higher than the model with $4$ slices.
Therefore, the optimum number of slices to use was taken as $4$. Additionally,
a more generalized observation shows that the model performance deteriorates
as more slices are used. Table 1 shows the incremental improvement of the base
model over a series of critical settings that contributed to a significant
boost in the model’s performance.

Table 1: Incremental improvement in model performance with different design
choices

Setting | Balanced Accuracy | Cohen’s Kappa | Lin’s CCC
---|---|---|---
Base Model | $75.5\pm 0.3$ | $69.0\pm 0.6$ | $85.1\pm 0.3$
With Consensus Grading | $77.6\pm 0.3$ | $69.6\pm 0.3$ | $85.8\pm 0.4$
With Imagenet weights | $78.8\pm 1.3$ | $70.7\pm 0.6$ | $86.8\pm 0.5$
With Augmentations | $81.1\pm 0.8$ | $72.6\pm 0.6$ | $88.1\pm 0.4$
With TTA | $\mathbf{81.7\pm 0.6}$ | $\mathbf{73.8\pm 0.4}$ | $\mathbf{88.5\pm 0.1}$

####  3.2.3 Spine grading: Comparison to baselines

We compared our work with the current state-of-the-art methods that quantified
LDD using PG. To ensure a fair comparison with our work, all the models are
trained on the NFBC dataset with the weights initialization, hyperparameters,
and loss functions prescribed by the authors in their work. While SpineNetV1
and SpineNetV2 are trained with their proposed architectures, we modified
Context-Aware Transformers for our baseline by implementing the encoder part
of the original network. This is due to the limitation on data availability
for MRI sequences used in their method. A comparison of these methods with our
work is shown in Table 2. Additionally, we extended the comparison beyond BA
to measure the model performance at different subgroups as shown in Fig. 4.
Compared to our work, the prior works do not generalize well on our dataset
when assessed on IVD and PG levels.

Table 2: Benchmark performance of our model against state-of-the-art spine
grading models

Model Architecture | Balanced Acc | Cohen’s Kappa | Lin’s CCC
---|---|---|---
SpineNetV1 | $74.3\pm 0.6$ | $63.7\pm 1.2$ | $83.4\pm 0.7$
SpineNetV2 | $77.5\pm 0.5$ | $67.2\pm 0.9$ | $85.0\pm 0.6$
SCT-Encoder | $78.2\pm 0.8$ | $69.6\pm 0.9$ | $86.4\pm 0.5$
Ours | $\mathbf{81.7\pm 0.6}$ | $\mathbf{73.8\pm 0.4}$ | $\mathbf{88.5\pm 0.1}$

Fig. 4: Inter-rater reliability and BA of our model compared to other methods
at more granular levels of Sex, IVD, and PGs

##  4 Discussion

In this work, we revisited different architectures used for LDD assessment
using the PG system and proposed a robust baseline that outperformed existing
methods. An emphasis is placed on the importance of design choices and
settings contributing to improved model performance. Our model exhibits more
generalized characteristics and does not bias towards a specific subgroup, as
observed in other methods. We believe that our study will contribute towards
better robust stratification of patients for LDD using lumbar spine MRIs. Our
code and pre-trained models are released at https://oulu-
imeds.github.io/ISBI2023_PfirrmannGrading/ to advance further studies.

##  5 Compliance with ethical standards

NFBC1966 participants gave their informed consent and the Regional Ethics
Committee of the Northern Ostrobothnia District approved the study.

##  6 Acknowledgements

This work was funded by the Marie Sklodowska-Curie Actions (MCSA),
International Training Network, under grant agreement 955735 for the Disc4All
consortium. This work was also supported by funding from the Academy of
Finland (Profi6 336449 funding program).

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