# Multifractal eigenfunctions
for a singular quantum billiard

Jonathan P. Keating  Mathematical Institute, University of Oxford, Andrew
Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG,
UK<EMAIL_ADDRESS>and  Henrik Ueberschär  Sorbonne Université
and Université de Paris, CNRS, IMJ-PRG, F-75006 Paris.
<EMAIL_ADDRESS>
###### Abstract.

Whereas much work in the mathematical literature on quantum chaos has focused
on phenomena such as quantum ergodicity and scarring, relatively little is
known at the rigorous level about the existence of eigenfunctions whose
morphology is more complex.

Quantum systems whose dynamics is intermediate between certain regimes – for
example, at the transition between Anderson localized and delocalized
eigenfunctions, or in systems whose classical dynamics is intermediate between
integrability and chaos – have been conjectured in the physics literature to
have eigenfunctions exhibiting multifractal, self-similar structure. To-date,
no rigorous mathematical results have been obtained about systems of this kind
in the context of quantum chaos.

We give here the first rigorous proof of the existence of multifractal
eigenfunctions for a widely studied class of intermediate quantum systems.
Specifically, we derive an analytical formula for the Renyi entropy associated
with the eigenfunctions of arithmetic S̆eba billiards, in the semiclassical
limit, as the associated eigenvalues tend to infinity.

We also prove multifractality of the ground state for more general, non-
arithmetic billiards and show that the fractal exponent in this regime
satisfies a symmetry relation, similar to the one predicted in the physics
literature, by establishing a connection with the functional equation for
Epstein’s zeta function.

##  1\. Introduction

Following the proof of the quantum ergodicity theorem by Snirelman, Zelditch
and Colin de Verdière [38, 41, 13], one of the central questions in the
quantum chaos literature over the past fourty years has been the
classification of semiclassical measures which may arise in relation to the
eigenfunctions of quantized chaotic systems. Rudnick and Sarnak proposed their
Quantum Unique Ergodicity (QUE) conjecture in 1994 [30], asserting that the
unique semiclassical measure which should arise along the sequence of Laplace
eigenfunctions on manifolds with negative curvature is the Liouville measure.
This conjecture was proved by Lindenstrauss and Soundararajan [25, 39] in the
case of arithmetic hyperbolic surfaces. Holowinsky and Soundararajan proved a
holomorphic analogue of QUE for Hecke eigenforms [17, 18]. Anantharaman [2]
generalized this work by ruling out the existence of semiclassical measures
localized on a union of points and geodesic segments for Anosov manifolds. On
the other hand, Hassell showed that the stadium billiard, although ergodic,
does not satisfy QUE [15]. The existence of scars was proved by Faure,
Nonnenmacher and de Bièvre for hyperbolic toral automorphisms of minimal
periods [14].

Another class of systems which has attracted considerable attention in the
quantum chaos literature are those termed pseudointegrable [29]. Such systems
are classically close to integrable, yet their quantum dynamics displays
features similar to those found in chaotic systems. In particular, their
dynamics in phase space is not constrained to tori, but more generally to
handled spheres.

Examples of pseudointegrable billiards are rational polygonal billiards. More
generally, one sometimes includes systems which can be understood as toy
models of pseudointegrable systems, such as quantum star graphs [19],
parabolic toral automorphisms [26] and S̆eba billiards [34]. The latter is a
rectangular billiard with a Dirac mass placed in its interior. This system can
be rigorously studied by means of von Neumann’s theory of self-adjoint
extensions. Over the past twenty years much work has been devoted to the study
of the spectrum and eigenfunctions of such billiards [12, 5, 6, 7, 8, 19, 31,
23, 24, 21, 22]. In particular, arithmetic S̆eba billiards have been shown to
be quantum ergodic [31, 23], whereas diophantine S̆eba billiards possess
scarred semiclassical measures [24, 21].

While phenomena such as quantum ergodicity and scarring have attracted a good
deal of attention in the mathematical literature on quantum chaos, it is
important to point out that many quantum systems possess eigenfunctions whose
morphology is far more complex than is simply characterised by their being
purely localized or ergodic. For example, in a wide range of quantum systems
in which the spectral statistics sit at the transition between Poisson and
random-matrix, it has been argued that the eigenfunctions should exhibit
multifractal self-similar structure. Pseudointegrable systems are one such
class of examples, because their dynamics is intermediate between
integrability and chaos.

Multifractality is a phenomenon which is observed in many scientific fields
ranging from financial mathematics and biology to quantum physics. It
describes systems whose self-similarity in a certain scaling regime is so
complex that it cannot be described by a single fractal exponent but only by a
spectrum of exponents. The multifractality of eigenfunctions is one of the
most important phenomena in the theoretical and experimental study of
intermediate quantum systems. Physicists have conjectured its existence in a
large of class of systems such as the Anderson model [1, 33, 28],
pseudointegrable systems [9], quantum spin chains [3], quantum maps [27] and
in certain ensembles of random matrices [20, 11], based on heuristic arguments
and extensive numerical investigations. Moreover, they have conjectured that
the fractal exponents $D_{q}$ should satisfy a symmetry relation [27] with
respect to the critical parameter value $q=\frac{1}{4}$. For an introduction
to the extensive physics literature on this subject, see [4].

Although multifractality is an extremely active field in physics, and
theoretical evidence for the multifractality of quantum eigenfunctions has
been provided by many works in the physics literature as well as physical and
numerical experiments, no rigorous mathematical proof is known to date for any
class of intermediate systems.

In this paper we prove the multifractality of the eigenfunctions for S̆eba’s
billiard in the strong coupling regime. We derive an analytical formula for
the fractal exponents $D_{q}$. We also prove multifractality of the ground
state and show that the symmetry relation of the fractal exponents around the
critical exponent $q=\frac{1}{4}$ is a consequence of the functional equation
of the Epstein zeta function.

Acknowledgements: This work was accomplished during several visits to the
Mathematical Institute at the University of Oxford. H.U. would like to thank
the institute and particularly Jon Keating’s group for their kind hospitality
during his stay. The authors would like to thank Nalini Anantharaman, Jens
Marklof and Zeev Rudnick for many helpful comments which led to the
improvement of this paper.

H.U. gratefully acknowledges the support of the CNRS by means of a “délégation
CNRS”. In addition, H.U. was supported by the grant ANR-17-CE40-0011-01 of the
French National Research Agency ANR (project SpInQS). JPK is pleased to
acknowledge support from ERC Advanced Grant 740900 (LogCorRM).

##  2\. Background and statement of the results

###  2.1. Mathematical formulation of the problem

Denote by $(X,\mu)$ a measure space. In classical multifractal analysis one
usually evaluates $\mu$ on a box partition ${\mathcal{B}}_{r}$ of $X$, where
$r$ denotes the side length of the boxes. To characterise the self-similarity
of the measure $\mu$ one studies the behaviour of the moment sum

$M_{q}(r)=\sum_{B\in{\mathcal{B}}_{r}}\mu(B)^{q}$

in the limit as $r\to 0$. One expects a scaling law of the form $M_{q}(r)\sim
r^{D_{q}}$, where $D_{q}$ denotes the fractal exponent. A key goal of
multifractal analysis is to determine $D_{q}$ as a function of $q$.

Let us consider the Laplace-Beltrami operator $-\Delta$ on a two-dimensional
Euclidean compact manifold ${\mathcal{M}}$ with or without boundary. An
eigenfunction $\psi_{\lambda}$ of $-\Delta$, satisfying
$(\Delta+\lambda)\psi_{\lambda}=0$, may be represented as a superposition of
plane waves

(2.1)
$\psi_{\lambda}(x)=\sum_{\xi\in{\mathcal{L}}}\hat{\psi}_{\lambda}(\xi)e^{i\xi\cdot
x}.$

This may be achieved by embedding the Euclidean billiard in a rectangular
enclosure ${\mathcal{R}}$ and representing $\psi_{\lambda}$ with respect to
the canonical orthonormal basis of complex exponentials $e^{i\xi\cdot x}$,
where $\xi\in{\mathcal{L}}$ and ${\mathcal{L}}$ is a rectangular lattice. We
note that the choice of rectangle, and therefore of the lattice
${\mathcal{L}}$, ought to translate into a scaling factor which will appear in
lower order terms, because, in order to calculate the fractal exponent
$D_{q}$, one considers the asymptotic behaviour of $\log M_{q}(r)$. A rigorous
formulation of the problem of multifractal eigenfunctions on Euclidean
billiards is beyond the scope of this introduction and will be addressed in a
separate article. We stress that for the case of S̆eba billiards no such
ambiguity with respect to the choice of lattice exists.

Let us introduce the probability measure

$\mu_{\lambda}(\xi)=|\hat{\psi}_{\lambda}(\xi)|^{2}$

where we assume that $\psi$ is $L^{2}$-normalized.

We consider the moment sum

(2.2) $M_{q}(\lambda)=\sum_{\xi\in{\mathcal{L}}}\mu_{\lambda}(\xi)^{q}.$

To understand how to determine the scaling parameter which plays the role of
$r$, one must distinguish different regimes. We will focus here first on the
semiclassical regime ($\lambda\to+\infty$). However, one may also study
multifractality of the ground state ($\lambda\to 0$), which we address in
section 2.5.

###  2.2. Multifractality in the semiclassical regime

To motivate the definition of the fractal exponent associated with the moment
sum (2.2), let us first of all consider the case of a finite-dimensional
Hilbert space. Each eigenstate may be represented by a vector
$v\in{\mathbb{C}}^{d}$, where $d$ denotes the dimension of the space. Let us
normalize such that the largest coordinate equals $1$ ($\ell^{\infty}$
normalization).

The fractal exponent $D_{q}$ describes how the moment sum associated with $v$
scales with respect to the dimension $d$:

$\sum_{k=1}^{d}v_{k}^{2q}\sim d^{D_{q}}.$

If $v$ is flat, then all entries equal $1$ so that $D_{q}=1$. On the other
hand, if $v$ is maximally localized, meaning that one entry equals $1$ and all
others vanish, then $D_{q}=0$.

Often one normalizes with respect to the $\ell^{2}$-norm. In this case one
observes a decay of the type $\sim d^{(1-q)D_{q}}$. The fractal exponent
$D_{q}$ may be defined in terms of the Renyi entropy of the probability
measure $p_{i}=v_{i}^{2}/\|v\|^{2}$ in the following way:

$D_{q}=\frac{H_{q}(v)}{\log d},\quad
H_{q}(v)=\frac{1}{1-q}\log\left(\sum_{i=1}^{d}p_{i}^{q}\right).$

Let us generalize this definition to an infinite-dimensional Hilbert space.
The measure $\mu_{\lambda}(\xi)$, which we introduced above, must carry most
of its mass when $|\xi|^{2}$, the eigenvalue of the plane wave, is close to
$\lambda$. Geometrically this means that the measure $\mu_{\lambda}$ will
concentrate most of its mass inside an annulus of central radius
$\sqrt{\lambda}$.

Let us consider the case of S̆eba’s billiard (cf. section 3 for an
introduction), a rectangular billiard with a Dirac mass placed in its
interior. The spectrum consists of two parts: old Laplace eigenvalues whose
eigenfunctions do not feel the presence of the scatterer, and new eigenvalues
whose eigenfunctions feel its presence. We will study the multifractal
properties of these new eigenfunctions.

Let us denote the set of old Laplace eigenvalues (counted without
multiplicity) by

${\mathcal{N}}:=\\{0=n_{0}<n_{1}<\dots<n_{j}<\cdots\to+\infty\\}.$

The set $\Lambda=\\{\lambda_{j}\\}_{j\geq 0}$ of new eigenvalues forms an
interlacing sequence with $(n_{j})_{j\geq 0}$. We denote by
$\Delta_{j}=\min_{m\in{\mathcal{N}}}|m-\lambda_{j}|=|\tilde{n}_{j}-\lambda_{j}|$
the distance of a new eigenvalue $\lambda_{j}$ to the closest neighbouring
Laplace eigenvalue, which we denote by $\tilde{n}_{j}$. If two such
eigenvalues exist we denote by $\tilde{n}_{j}$ the smaller of the two.

Moreover, let us introduce the mean distance

$\left\langle\Delta_{j}\right\rangle_{x}=\frac{1}{\\#\\{k\mid\lambda_{k}\leq
x\\}}\sum_{\lambda_{k}\leq x}\Delta_{k}.$

We distinguish two quantizations of S̆eba’s billiard:

  * (i)

weak coupling: $\left\langle\Delta_{j}\right\rangle_{x}=O((\log x)^{-1/2})$

  * (ii)

strong coupling: $\left\langle\Delta_{j}\right\rangle_{x}=(\log
x)^{\alpha+o(1)}$, $\alpha=\alpha(\Lambda)\in(-\tfrac{1}{2},\tfrac{1}{2}]$.

Whereas the eigenvalues of the weak coupling quantization arise from self-
adjoint extension theory, a renormalization of the extension parameter gives
rise to the strong coupling quantization condition. For any given
$\alpha\in(-\tfrac{1}{2},\tfrac{1}{2}]$, there exists a suitable
renormalization to render the corresponding quantization. We refer the reader
to section 3 for a detailed discussion.

######  Remark 1.

Note, if $\alpha(\Lambda)<\tfrac{1}{2}$, this does not preclude the mean
spacing $\left\langle\delta_{j}\right\rangle_{x}$, where
$\delta_{j}=n_{j}-\lambda_{j}$, from being of the same order as the mean
spacing of the Laplace eigenvalues:
$\left\langle\delta_{j}\right\rangle_{x}\asymp\sqrt{\log x}$111Note that the
mean spacing of the Laplace eigenvalues $n_{j}$ is of order $\sqrt{\log x}$ as
a consequence of Landau’s Theorem on the representation of integers as a sum
of two squares which states that $\\#\\{n_{j}\leq X\\}\sim Bx/\sqrt{\log x}$..
It simply means that the locations of the new eigenvalues fluctuate rather
than remaining close to the intermediate values
$\bar{n}_{j}=\tfrac{1}{2}(n_{j+1}+n_{j})$.

Let us return to the moment sum (2.2). In the case of S̆eba’s billiard we have
the following explicit formula for the probability measure $\mu_{\lambda}$ on
the lattice ${\mathbb{Z}}^{2}$:

$\mu_{\lambda}(\xi)=(|\xi|^{2}-\lambda)^{-2}\left(\sum_{\xi\in{\mathbb{Z}}^{2}}(|\xi|^{2}-\lambda)^{-2}\right)^{-1}$

which allows for a precise analysis.

Firstly, let us determine the scaling parameter. Since we are dealing with an
infinite-dimensional space, the dimension must be replaced with the number of
important components $N$ in the sum (2.2). $N$ will be defined in terms of the
average number of lattice points in the annulus which supports most of the
mass of the measure $\mu_{\lambda}$; in particular $N$ will depend on
$\lambda$.

In the case of S̆eba’s billiard, the moment sum (2.2) is of the form

(2.3)
$M_{q}(\lambda_{j})=\frac{\zeta_{\lambda_{j}}(2q)}{\zeta_{\lambda_{j}}(2)^{q}}$

where we introduce the zeta function

(2.4)
$\zeta_{\lambda}(s)=\sum_{m\in{\mathcal{N}}}r_{Q}(m)|m-\lambda|^{-s},\quad\operatorname{Re}s>1,$

and $r_{Q}(n)$ denotes the representation number of the quadratic form defined
by $Q(\xi_{1},\xi_{2})=|\xi|^{2}$ for $\xi=(\xi_{1},\xi_{2})\in{\mathcal{L}}$,
whereas ${\mathcal{N}}$ denotes the set of numbers which is representable as a
value of $Q$ for some $\xi\in{\mathcal{L}}$. In the following we will consider
the arithmetic case, when ${\mathcal{L}}={\mathbb{Z}}^{2}$ and
$r_{Q}(n)=r_{2}(n)$.

###  2.3. A first example: weak coupling

In the case of the weak coupling quantization we have
$\left\langle\Delta_{j}\right\rangle_{n}=O(1/\sqrt{\log n})$. In section 5.2
we will show that for a generic new eigenvalue only the lattice points on a
circle of radius $\sqrt{n}$ contribute, i.e. the number of important terms in
the sum (2.2) is $r_{2}(n)$.

In fact, the moment sum can be rewritten as

$M_{q}(\lambda_{j})=\frac{\zeta_{\lambda_{j}}(2q)}{\zeta_{\lambda_{j}}(2)^{q}}=\frac{m_{q}(\lambda_{j})}{m_{1}(\lambda_{j})^{q}}$

where

(2.5)
$m_{q}(\lambda_{j})=r_{2}(\tilde{n}_{j})+\Delta_{j}^{2q}\sum_{m\neq\tilde{n}_{j}}r_{2}(m)|m-\lambda_{j}|^{-2q}$

and, because $\Delta_{j}$ is typically small, only the term
$r_{2}(\tilde{n}_{j})$ contributes in the numerator and denominator
respectively. In particular, we will show that, along a full density
subsequence of eigenvalues,

$M_{q}(\lambda_{j})\sim r_{2}(\tilde{n}_{j})^{1-q}.$

For a full density subsequence of $n\in{\mathcal{N}}$ we have

$r_{2}(n)=(\log n)^{\tfrac{1}{2}\log 2+o(1)}$

because $r_{2}(n)$ is close to its logarithmic normal order. Therefore, we
define the scaling parameter as the logarithmic normal order of
$r_{2}(\tilde{n}_{j})$:

$N=(\log\tilde{n}_{j})^{\tfrac{1}{2}\log 2}$

We define the semiclassical fractal exponent, associated with a sequence of
eigenvalues, as follows.

######  Definition 2.1.

Let $q>\tfrac{1}{2}$. Let $\Lambda$ be a sequence of eigenvalues which
accumulates at infinity. We define the general entropy associated with the
eigenfunction $\psi_{\lambda}$ as

(2.6) $h_{q}(\psi_{\lambda})=\log m_{q}(\lambda).$

For $q\neq 1$ the Renyi entropy associated with the probability measure
$\mu_{\lambda}$ on the lattice ${\mathcal{L}}$ may then be defined as

(2.7)
$H_{q}(\mu_{\lambda})=\frac{h_{q}(\psi_{\lambda})-qh_{1}(\psi_{\lambda})}{1-q}=\frac{1}{1-q}\log\left(\sum_{\xi\in{\mathcal{L}}}\mu_{\lambda}(\xi)^{q}\right).$

The fractal exponents associated with $\Lambda$ are defined as

(2.8) $\displaystyle d_{q}^{\Lambda}=$
$\displaystyle\limsup_{\lambda\in\Lambda\to+\infty}\frac{h_{q}(\psi_{\lambda})}{\log
N}\quad\text{with respect to the $\ell^{\infty}$-normalization,}$ (2.9)
$\displaystyle D_{q}^{\Lambda}=$
$\displaystyle\limsup_{\lambda\in\Lambda\to+\infty}\frac{H_{q}(\mu_{\lambda})}{\log
N}\quad\text{with respect to the $\ell^{2}$-normalization.}$

######  Remark 2.

We point out that the Renyi entropy of a probability measure on a lattice
${\mathcal{L}}$ is a natural generalisation of the Shannon entropy familiar
from information theory [35, 36]. For a probability measure $p=p_{\xi}$ on a
lattice ${\mathcal{L}}$, the Shannon entropy is defined as follows

$H_{Sh.}(p)=-\sum_{\xi\in{\mathcal{L}}}p_{\xi}\log p_{\xi}.$

In fact, we have

$\displaystyle\lim_{q\to 1}H_{q}(p)$ $\displaystyle=\lim_{q\to
1}\frac{1}{1-q}\log\left(\sum_{\xi\in{\mathcal{L}}}p_{\xi}^{q}\right)$
$\displaystyle=-\left[\frac{d}{dq}\log\left(\sum_{\xi\in{\mathcal{L}}}p_{\xi}^{q}\right)\right]_{q=1}$
$\displaystyle=-\sum_{\xi\in{\mathcal{L}}}p_{\xi}\log p_{\xi}.$

So the Shannon entropy of $p$ is recovered from the Renyi entropy in the
limit, as $q\to 1$.

We have the following theorem which we prove in section 5.2. Due to the
weakness of the perturbation, the new eigenvalue is on average close to a
neighbouring Laplace eigenvalue. As a result we only have a single fractal
exponent.

######  Theorem 2.2 (A simple scaling law).

Let $\Lambda_{w}$ denote the sequence of new eigenvalues in the weak coupling
regime. There exists a full density subsequence
$\Lambda^{\prime}\subset\Lambda_{w}$ such that
$d_{q}(\Lambda^{\prime})=D_{q}(\Lambda^{\prime})=1$.

###  2.4. A multifractal regime: strong coupling

To determine the scaling parameter $N$ we have to calculate the number of
important terms in the moment sum (2.2). This is closely related to the number
of lattice points in an annulus

$A(\lambda,G)=\\{v\in{\mathbb{R}}^{2}\mid||\xi|^{2}-\lambda|\leq G\\},$

where $G=G(\lambda)$ is just large enough such that the terms corresponding to
lattice points outside the annulus $A(\lambda,G)$ are negligible, when
$\lambda\to+\infty$, and $G=o(\lambda)$. In particular, $G$ must grow faster
than the mean spacing of the Laplace eigenvalues:
$G/\sqrt{\log\lambda}\to+\infty$

Let us introduce the tail of the zeta function $\zeta_{\lambda}$:

$\tau_{q}(\lambda,G)=\sum_{\begin{subarray}{c}m\in{\mathcal{N}}\\\
|m-\lambda|\geq G\end{subarray}}r_{2}(m)|m-\lambda|^{-2q}.$

######  Definition 2.3.

We define the essential support of the measure $\mu_{\lambda}$ as the set of
lattice points $A(\lambda,G)\cap{\mathbb{Z}}^{2}$, where $G=G(\lambda)$
satisfies $G/\sqrt{\log\lambda}\to+\infty$, and is determined from the
condition

(2.10)
$\left\langle\Delta_{j}\right\rangle_{\lambda}^{2q}\left\langle\tau_{q}\right\rangle_{\lambda}=1$

and the mean tail is defined as

(2.11)
$\left\langle\tau_{q}\right\rangle_{T}=\frac{1}{T}\int_{T}^{2T}\tau_{q}(t,G)dt.$

######  Remark 3.

We will see below that $\left\langle\tau_{q}\right\rangle\asymp G^{1-2q}$. So
if we recall
$\left\langle\Delta_{j}\right\rangle_{\lambda}=(\log\lambda)^{\alpha+o(1)}$,
then we see that the definition of $G$ as a solution of (2.10) is quite robust
when $\lambda\to+\infty$. Strictly speaking, for the terms outside the annulus
$A(\lambda,G)$ to be negligible, one should require the r.h.s. of equation
(2.10) to be $o(1)$. However, for any function which decays sufficiently
slowly ($1/o(\log\lambda)$ to be precise) this will only change a lower order
term in the asymptotic of $\log G$. Because, as we will see below, it is only
the leading term in the asymptotic of $\log G$ which is relevant in the
definition of $D_{q}$, we set the r.h.s. of (2.10) equal to $1$ for
simplicity.

As the number of lattice points in a thin annulus fluctuates a lot, we define
the scaling parameter $N$ as the average number of lattice points in the
annulus $A(\lambda,G)$ which is given by its area.

######  Definition 2.4.

We define the scaling parameter as $N=\operatorname{vol}(A(\lambda,G))=2\pi
G$.

We have the following proposition which we prove in section 8.

######  Proposition 2.5.

Suppose $q>1/2$. We have

$\left\langle\tau_{q}\right\rangle_{\lambda}\sim\frac{2\pi}{2q-1}G^{1-2q}$

as $\lambda\to\infty$.

Our main result concerns the strong coupling regime. In this regime the
spacing of the new eigenvalues may be as large as the mean spacing of the
Laplace eigenvalues. We determine under which conditions on the average
distance of the new eigenvalues to the nearest Laplace eigenvalue a spectrum
of fractal exponents emerges. Our result includes the case of a quantization
consistent with level repulsion, provided the fluctuations around the mean
location of the new eigenvalues are large enough. We prove the following
theorem in section 5.1.

######  Theorem 2.6 (A multifractal scaling law).

Let $\Lambda_{s}$ denote a sequence of new eigenvalues such that
$\alpha=\alpha(\Lambda_{s})\in(\tfrac{1}{4},\tfrac{1}{2})$. There exists a
full density subsequence $\Lambda^{\prime}\subset\Lambda_{s}$ such that for
any $q>1$ which satisfies $(1-\log 2)(2-4\alpha)^{-1}<q\leq(2-4\alpha)^{-1}$
we have the following formulae, which may be analytically continued to the
complex plane,

(2.12)
$d_{q}(\Lambda^{\prime})=\frac{1}{2\alpha}\left(1-\frac{1}{2q}\right)\log 2$

and for a constant $c\in[\tfrac{1}{2}\log 2,1]$

(2.13)
$D_{q}(\Lambda^{\prime})=\frac{1}{2\alpha}\left(1-\frac{1}{2q}\right)\frac{2cq-\log
2}{q-1}.$

######  Remark 4.

It is important to remark that “rigid” quantizations which have the property
that the new eigenvalues stay close to the intermediate values of the Laplace
eigenvalues, in the sense that
$\left\langle\delta_{j}\right\rangle_{x}\asymp\left\langle\Delta_{j}\right\rangle_{x}=(\log
x)^{1/2+o(1)}$, may lead to a possible breakdown of multifractality. This is
interesting, because it suggests that in a regime, where level repulsion
occurs, we also require the fluctuations of the new eigenvalues around their
mean location to be large enough for multifractality to manifest itself. We
discuss this case in detail in section 6.

###  2.5. Multifractality of the ground state

Instead of considering the semiclassical regime, where $\lambda\to+\infty$,
the regime where $\lambda\to 0$ is also frequently studied. Physically this
corresponds to the ground state in a weak coupling regime. In this regime we
expect multifractality of the fluctuations of the eigenfunction around the
constant term in the Fourier series (2.1), as $\lambda\to 0$. This leads to
the study of the modified moment sums

(2.14) $M_{q}^{*}(\lambda)=\zeta^{*}_{\lambda}(2q)$

where

(2.15)
$\zeta_{\lambda}^{*}(s)=\sum_{n\in{\mathcal{N}}\setminus\\{0\\}}r_{Q}(n)|n-\lambda|^{-s},\quad\operatorname{Re}s>1.$

The parameter $N$ does not depend on $\lambda$ in this regime and, therefore,
we remove it from the definition of the fractal exponent.

######  Definition 2.7.

Let $q>1/2$. We define the fractal exponent as follows

$d_{q}^{*}=\lim_{\lambda\to 0}\log\zeta_{\lambda}^{*}(2q)$

and

$D_{q}^{*}=\frac{d_{q}^{*}-qd_{1}^{*}}{1-q}$

We state the second main result of this paper which provides an explicit
formula for $D_{q}^{*}$ in terms of Epstein’s zeta function

$\zeta_{Q}(s)=\sum_{(m,n)\neq(0,0)}Q(m,n)^{-s},\quad\operatorname{Re}s>1$

which is known to satisfy the functional equation

$\zeta_{Q}(s)=\varphi_{Q}(s)\zeta_{Q}(1-s).$

Moreover, we show that $D_{q}^{*}$ satisfies a symmetry relation with respect
to $q=\tfrac{1}{4}$ which is closely related to the functional equation of
Epstein’s zeta function. The following theorem is proved in section 7.

######  Theorem 2.8.

The fractal exponent $D^{*}_{q}$ admits an analytic continuation to the full
complex plane in $q$. It satisfies the following symmetry relation with
respect to the parameter value $q=\tfrac{1}{4}$:

$D_{1/2-q}^{*}=\frac{1-q}{\tfrac{1}{2}+q}\left(D_{q}^{*}+\frac{\log\varphi_{Q}(2q)+(2q-\tfrac{1}{2})\log\zeta_{Q}(2)}{1-q}\right).$

In particular, in the limit as $q\to 1$, the fractal exponent converges to the
Shannon entropy of the measure
$\mu_{\lambda}(\xi)=\zeta_{Q}(2)^{-1}|\xi|^{-4}$:

$\lim_{q\to
1}D_{q}^{*}=\log\zeta_{Q}(2)-2\frac{\zeta_{Q}^{\prime}(2)}{\zeta_{Q}(2)}$

which we may rewrite as

$H_{Sh.}(\mu_{\lambda})=-\sum_{\xi\in{\mathbb{Z}}^{2}}\frac{1}{\zeta_{Q}(2)|\xi|^{4}}\log\left(\frac{1}{\zeta_{Q}(2)|\xi|^{4}}\right).$

##  3\. Background on S̆eba billiards

###  3.1. Spectral theory of S̆eba billiards

S̆eba billiards belong to the class of pseudointegrable billiards and over the
past 30 years they have been studied extensively in the literature. For a
detailed discussion of the spectral theory of such billiards we refer the
reader to the introductory sections of the articles [31] and [40].

In this article we will deal with an arithmetic S̆eba billiard associated with
the square lattice ${\mathbb{Z}}^{2}$. Let
${\mathbb{T}}^{2}={\mathbb{R}}^{2}/2\pi{\mathbb{Z}}^{2}$ and
$x_{0}\in{\mathbb{T}}^{2}$. The spectrum of the positive Laplacian
$-\Delta=-(\partial_{x}^{2}+\partial_{y}^{2})$ is given by integers which may
be represented as a sum of two squares $n=x^{2}+y^{2}$,
$(x,y)\in{\mathbb{Z}}^{2}$. We denote the ordered set of such numbers as
${\mathcal{N}}$. The multiplicities in the Laplace spectrum on
${\mathbb{T}}^{2}$ are, thus, given by the representation number

$r_{2}(n)=\\#\\{(x,y)\in{\mathbb{Z}}^{2}\mid n=x^{2}+y^{2}\\}$

which is an arithmetic function that has been extensively studied in the
number theory literature.

Consider a self-adjoint extension $-\Delta_{\varphi}$, $\varphi\in[-\pi,\pi)$,
of $-\Delta$ restricted to the domain
$C_{c}^{\infty}({\mathbb{T}}^{2}\setminus\\{x_{0}\\})$. Physically this
operator corresponds to the weak coupling quantization for a square torus with
a Dirac delta potential (as introduced by S̆eba in [34]).

The spectrum of the operator $-\Delta_{\varphi}$ consists of two parts:

  * (i)

old (Laplace) eigenvalues with multiplicity reduced by $1$

  * (ii)

new eigenvalues with multiplicity $1$ which interlace with the Laplace
eigenvalues

The eigenspace associated with an old eigenvalues is simply the codimension
one subspace of Laplace eigenfunctions which vanish at $x_{0}$. The old
eigenfunctions do not feel the presence of the potential. In this article, we
will only be interested in the new eigenfunctions which do feel the presence
of the potential.

These new eigenfunctions are Green’s functions associated with the resolvent
of the Laplacian, where one variable has been fixed as the position of the
potential $x_{0}$. Because of translation invariance we may set $x_{0}=0$. We
then have the following $L^{2}$-representation of the new eigenfunctions

(3.1) $\begin{split}G_{\lambda}(x)\stackrel{{\scriptstyle
L^{2}}}{{=}}&\sum_{\xi\in{\mathbb{Z}}^{2}}\frac{e^{i\left\langle\xi,x\right\rangle}}{|\xi|^{2}-\lambda}\\\
=&\sum_{n\in{\mathcal{N}}}\frac{1}{n-\lambda}\sum_{|\xi|^{2}=n}e^{i\left\langle\xi,x\right\rangle}.\end{split}$

Moreover, the new eigenvalues may be determined as the solutions of the
following spectral equation

(3.2)
$\sum_{n\in{\mathcal{N}}}r_{2}(n)\left\\{\frac{1}{n-\lambda}-\frac{n}{n^{2}+1}\right\\}=c_{0}\tan(\frac{\varphi}{2}).$

Because the function on the l.h.s. of equation (3.2) is monotonous on
intervals $(n_{j},n_{j+1})$, $n_{j}\in{\mathcal{N}}$, we easily see that the
new eigenvalues form a sequence of real numbers which interlace with the
sequence of Laplace eigenvalues:

$\lambda_{0}<0=n_{0}<\lambda_{1}<n_{1}<\lambda_{2}<n_{2}<\cdots<\lambda_{j}<n_{j}<\cdots\to+\infty.$

An important question concerns the statistics of the spacings
$\delta_{j}:=n_{j}-\lambda_{j}$. As was shown in [32], $\lambda_{j}$ is on
average close to $n_{j}$, precisely $\delta_{j}=O((\log n_{j})^{-1/2})$ for a
full density subsequence of Laplace eigenvalues. This shows that the
quantization procedure using self-adjoint extension theory, which was
introduced by S̆eba [34], gives rise to a so-called weak coupling
quantization. Shigehara [37] and later Bogomolny, Gerland and Schmit [10]
explained that a semiclassical renormalization of the extension parameter
$\varphi$ was necessary in order to obtain a strong coupling quantization
giving rise to a spectrum and eigenfunctions which are expected to have
interesting features.

###  3.2. Renormalization and strong coupling quantizations

In fact, a logarithmic renormalization of the coupling parameter,
$\tan(\varphi_{\lambda}/2)\sim c\log\lambda$, is equivalent to a truncation of
the sum on the l.h.s. of equation (3.2) outside a window of size
$\lambda^{1/2}$. We refer the reader to [40] for a detailed discussion of this
renormalization procedure. The reason for this equivalence is that the tail of
the sum has a logarithmic asymptotic as $\lambda\to+\infty$ which is cancelled
by the leading term in the asymptotic of $\tan(\varphi_{\lambda}/2)$. The
second order term in this asymptotic is related to local information about the
position of the new eigenvalues.

We thus obtain the following strong coupling quantization, where the function
$\beta=\beta(\lambda)=o(\log\lambda)$ is related to the inverse coupling
strength of the potential, and $\lambda_{j}$ is determined as the solution on
the interval $(n_{j},n_{j+1})$ as the solution to the following equation

(3.3) $\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\ |n-n_{j}|\leq
n_{j}^{1/2}\end{subarray}}\frac{r_{2}(n)}{n-\lambda}=\beta(\lambda).$

The relationship between the precise form of $\beta$ and the location of the
new eigenvalues is not understood at the rigorous level. When $\beta$ is
constant, we expect the spacings $\delta_{j}=n_{j}-\lambda_{j}$ to be of the
order of the mean spacing of the Laplace eigenvalues,
$\left\langle\delta_{j}\right\rangle_{x}\sim c\sqrt{\log x}$. However, one may
choose $\beta=c(\log\lambda)^{b}$, $b\in(0,1)$, to achieve spacings of lower
order.

It is important to remark that the location of a new eigenvalue $\lambda_{j}$
within the interval $(n_{j},n_{j+1})$ depends on the interplay of the
coefficients $r_{2}(n)$ in the summation in equation (3.3). Due to the strong
fluctuations of the function $r_{2}(n)$ one may have
$\left\langle\delta_{j}\right\rangle_{x}\asymp\sqrt{\log x}$, even though
$\left\langle\Delta_{j}\right\rangle_{x}\ll(\log x)^{\alpha}$,
$\alpha<\tfrac{1}{2}$.

##  4\. An estimate for $\zeta_{\lambda_{j}}(2q)$

In this section we will prove the following lemma.

######  Lemma 4.1.

Let $q>1$. There exists a subsequence of full density
${\mathcal{N}}_{1}\subset{\mathcal{N}}$ such that for any
$m\in{\mathcal{N}}_{1}$ we have the bound

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\ n\neq
m\end{subarray}}\frac{r_{2}(n)}{|m-n|^{2q}}\ll(\log m)^{-q+1/2+o(1)}.$

###### Proof.

Let $m\in{\mathcal{N}}_{1}(x)$. For convenience denote $L=\log m$. We first of
all claim that there is a full density subsequence of
${\mathcal{N}}_{1}\subset{\mathcal{N}}$ such that for $m\in{\mathcal{N}}_{1}$
(proof to be given in a separate section)

  * (i)

if $r\in[L^{1/2+o(1)},L^{2}]$ there are $\gg r/((\log m)^{1/2+o(1)}(\log
r)^{2})$ elements in ${\mathcal{N}}$ that lie in between $m$ and $n$ if
$r_{2}(n)\geq r$

  * (ii)

if $r\geq L^{2}$ there are $\gg_{\epsilon}r^{5/4-\epsilon}/\log r$ elements in
${\mathcal{N}}$ that lie in between $m$ and $n$

  * (iii)

the nearest neighbours of $m$ are at distance at least $L^{1/2-o(1)}$

  * (iv)

for $G\in[2,m^{1-\epsilon}]$ we have the bound

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\ |m-n|\geq
G\end{subarray}}|m-n|^{-2q}\leq\frac{(\log G)^{2}}{G^{2q-1}L^{1/2-o(1)}}$

We next divide the sum in three parts according to the size of $r_{2}(n)$:

Small $r_{2}(n)$: $r_{2}(n)\leq L^{1/2+o(1)}$. We have

$\displaystyle\sum_{\begin{subarray}{c}|m-n|\gg L^{1/2-o(1)}\\\ r_{2}(n)\leq
L^{1/2+o(1)}\end{subarray}}\frac{r_{2}(n)}{|m-n|^{2q}}\ll$
$\displaystyle\frac{L^{1/2+o(1)}}{L^{q-1/2-o(1)}L^{1/2-o(1)}}=L^{-q+1/2+o(1)}$

Medium $r_{2}(n)$: Set $R_{i}=2^{i}(\log m)^{1/2+o(1)}$. By property (6) (cf.
[23]) we may set $T=|m-n|$ and we then obtain

$\frac{T(\log T)^{2}}{(\log m)^{1/2-o(1)}}\gg\frac{R_{i}}{L^{1/2+o(1)}(\log
R_{i})^{2}}$

which implies $T\gg R_{i}/L^{o(1)}$.

We thus have

(4.1)
$\begin{split}\sum_{r_{2}(n)\in[R_{i},R_{i+1}]}\frac{r_{2}(n)}{|m-n|^{2q}}&\ll\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\
|m-n|\gg R_{i}/L^{o(1)}\end{subarray}}\frac{R_{i}}{|m-n|^{2q}}\\\
&\ll\frac{R_{i}(\log R_{i})^{2}}{L^{1/2-o(1)}R_{i}^{2q-1}}\\\
&=\frac{i^{2}}{L^{q-1/2-o(1)}2^{i(2q-2)}}\end{split}$

and summing over $i\geq 0$ yields the result (we used $q>1)$.

Large $r_{2}(n)$: We imitate the argument used in the proof of property (8) in
[23].
Take $R_{i}=2^{i}L^{2}/100$. Suppose $r_{2}(n)\in[R_{i},2R_{i}]$. By property
(ii) we have $|m-n|\gg_{\epsilon}R_{i}^{5/4-\epsilon}$. And, thus, we have

$\sum_{r_{2}(n)\in[R_{i},2R_{i}]}\frac{r_{2}(n)}{|m-n|^{2q}}\ll
R_{i}\sum_{|m-n|\gg_{\epsilon}R_{i}^{5/4-\epsilon}}|m-n|^{-2q}\ll_{\epsilon}R_{i}(R_{i}^{5/4-\epsilon})^{1-2q}L^{-1/2+o(1)}$

and this is bounded by $2^{i(9/4-5q/2+\epsilon)}L^{4-5q+\epsilon}$.

Summing over $i$ (recall $q>1$, so $9/4-5q/2<0$) we obtain

$\sum_{r_{2}(n)\geq
L^{2}}\frac{r_{2}(n)}{|m-n|^{2q}}=O_{\epsilon}(L^{4-5q+\epsilon})=O(L^{1-2q})$

because $q>1$ implies $4-5q<1-2q$.

∎

###  4.1. Proof that ${\mathcal{N}}_{1}$ is of full density:

  * (i)

We use the same strategy as in the proof of property (7) in [23]. Given that
$n\in{\mathcal{N}}$ satisfies $r=r_{2}(n)\in[(\log n)^{1/2}\log\log n,(\log
n)^{2}]$ we remove
$2r/((\log n)^{1/2}(\log\log n)(\log r)^{2})$ elements of ${\mathcal{N}}$ to
the left and right of $n$ and denote the set of such removed elements by
$R_{n}$. As was shown in the proof of property (7), it is sufficient to bound
the total number of removed elements in all sets $R_{n}$ such that $n\leq 2x$.
If we show that this set is of zero density in ${\mathcal{N}}$ (i.e. its
cardinality is $o(x/\sqrt{\log x})$), then we have shown that the set of
remaining elements is of full density in ${\mathcal{N}}$. Moreover, for any
$m$ in this full density set and any $n\in{\mathcal{N}}$ such that
$r_{2}(n)\geq r$ and $r\in[(\log n)^{1/2}\log\log n,(\log n)^{2}]$ we have at
least $r/((\log n)^{1/2}(\log\log n)(\log r)^{2})$ elements of ${\mathcal{N}}$
between $m$ and $n$.

It remains to be shown that the union of $R_{n}$, $n\leq 2x$, is indeed of
cardinality $o(x/\sqrt{\log x})$. As in the proof of (7), it is sufficient (in
view of the upper bound $(\log n)^{2}$) to consider $n\in[x/(\log x)^{10},2x]$
and we, thus, have $\log n=(1+o(1))\log x$.

In view of the asymptotic

$\sum_{n\leq x}r_{2}(n)\sim\pi x$

Chebyshev gives that the number of $n$ such that $r_{2}(n)\geq T$ is $\ll
x/T$.

Let $r_{2}(n)\in[R_{i},2R_{i}]$ with $R_{i}=2^{i}(\log x)^{1/2}\log\log x$. So
the total number of removed elements is

$\ll\frac{x}{R_{i}}\frac{R_{i}}{(\log x)^{1/2}(\log\log x)(\log
R_{i})^{2}}\ll\frac{x}{(\log x)^{1/2}(\log\log x)i^{2}}$

and summing over $i$ shows that the total number of elements removed is
$o(x/\sqrt{\log x})$.

  * (ii)

We follow the same strategy as in the proof of property (8) in [23]. Given
$n\in{\mathcal{N}}$ such that $r=r_{2}(n)\geq(\log n)^{2}$ we remove
$2r^{5/4-\epsilon}/\log r$ elements of ${\mathcal{N}}$ to the left and right.
We use the bound

$\sum_{n\leq x}r_{2}(n)^{2}\ll x\log x.$

As in the proof of (8) it is sufficient to consider $n\in[\sqrt{x},2x]$. Let
$R_{i}=2^{i}(\log x)^{2}/100$ and consider $r\in[R_{i},2R_{i}]$. Chebyshev
gives that the number of $n\leq 2x$ s. t. $r_{2}(n)\geq R_{i}$ is
$\ll\frac{x\log x}{R_{i}^{2}}$ and the associated removed contribution is

$\ll\frac{x\log x}{R_{i}^{2}}\frac{R_{i}^{5/4-\epsilon}}{\log R_{i}}\ll x\log
x~{}R_{i}^{-3/4-\epsilon}=x2^{-i(3/4+\epsilon)}(\log x)^{-1/2-\epsilon}$

and summing over $i$ shows that the total number of removed elements is
$o(x/\sqrt{\log x})$.

  * (iii)

This is the same as property (3) in [23].

  * (iv)

The proof is analogous to the proof of property (9) in [23], where
$(m-n)^{-2}$ is replaced with $|m-n|^{-2q}$ and we use

$\sum_{h\geq T}\frac{c(h)}{h^{2q}}\ll T^{1-2q}.$

##  5\. Proofs of Theorems 2.6 and 2.2

Let us consider the moments

$\displaystyle m_{q}(\lambda_{j})=$
$\displaystyle\Delta_{j}^{2q}\sum_{\xi\in{\mathbb{Z}}^{2}}\frac{1}{||\xi|^{2}-\lambda|^{2q}}$
$\displaystyle=$
$\displaystyle(\tilde{n}_{j}-\lambda_{j})^{2q}\sum_{n\in{\mathcal{N}}}r_{2}(n)|n-\lambda_{j}|^{-2q},\quad
q>1,$

which give information about the multifractal properties of the
eigenfunctions.

###  5.1. Proof of Theorem 2.6

Let us denote

${\mathcal{N}}_{2}=\\{n\in{\mathcal{N}}_{1}\mid r_{2}(n)=(\log
n)^{\tfrac{1}{2}\log 2+o(1)}\\}$

which is of full density in ${\mathcal{N}}$ (cf. [23]).

Moreover, $\alpha(\Lambda)\in(0,\tfrac{1}{2})$ and Chebyshev imply that
$\Delta_{j}\ll(\log\tilde{n}_{j})^{\alpha+o(1)}$ is a full density condition
if we assume $(\log\tilde{n}_{j})^{o(1)}\nearrow+\infty$. We denote by
${\mathcal{N}}^{\prime}$ the subset of ${\mathcal{N}}_{2}$ which satifies this
condition.

We need to estimate the fractal exponent associated with the subsequence
${\mathcal{N}}^{\prime}$

$D_{q}=\frac{1}{1-q}\limsup_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}}\frac{\log
m_{q}(\tilde{n}_{j})-q\log m_{1}(\tilde{n}_{j})}{\log N}.$

Let $L=\log\tilde{n}_{j}$. For $q>1$ and
$\tilde{n}_{j}\in{\mathcal{N}}^{\prime}$ we have

(5.1)
$\begin{split}m_{q}(\tilde{n}_{j})=~{}&r_{2}(\tilde{n}_{j})+\Delta_{j}^{2q}O(L^{1/2-q+o(1)})\\\
=~{}&L^{\log 2/2+o(1)}+O(L^{2\alpha q+o(1)})\times
O(L^{1/2-q+o(1)})\end{split}$

where we used Lemma 4.1.

We have $q>(1-\log 2)/(2-4\alpha)$ which implies
$-(1-2\alpha)q+\tfrac{1}{2}<\tfrac{1}{2}\log 2$ so that we have the asymptotic

$m_{q}(\tilde{n}_{j})\sim~{}L^{\log 2/2+o(1)}.$

Thus, we find

$\limsup_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}}\frac{\log
m_{q}(\tilde{n}_{j})}{\log\log\tilde{n}_{j}}=\lim_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}}\frac{\log
m_{q}(\tilde{n}_{j})}{\log\log\tilde{n}_{j}}=\frac{1}{2}\log 2.$

We need a separate estimate for $m_{1}(\tilde{n}_{j})$, because the above
estimate assumes $q$ to be sufficiently large.

For $q=1$ we have

$\sum_{n\in{\mathcal{N}},\;n\neq\tilde{n}_{j}}\frac{r_{2}(n)}{(n-\tilde{n}_{j})^{2}}=\sum_{\begin{subarray}{c}n\in{\mathcal{N}},\;n\neq\tilde{n}_{j}\\\
r_{2}(n)\leq
L^{2}\end{subarray}}\frac{r_{2}(n)}{(n-\tilde{n}_{j})^{2}}+\sum_{\begin{subarray}{c}n\in{\mathcal{N}},\;n\neq\tilde{n}_{j}\\\
r_{2}(n)>L^{2}\end{subarray}}\frac{r_{2}(n)}{(n-\tilde{n}_{j})^{2}}$

where the second sum is bounded by $O_{\epsilon}(L^{-1+\epsilon})$ in view of
the bound obtained for sums over large $r_{2}(n)$ in the proof of Lemma 6.1,
and the first sum is bounded by

$L^{2}\sum_{|n-\tilde{n}_{j}|\gg L^{1/2-o(1)}}(n-\tilde{n}_{j})^{-2}\ll
L^{1+o(1)}$

where we used (9) in [23].

This yields the bound

(5.2)
$\sum_{n\in{\mathcal{N}},\;n\neq\tilde{n}_{j}}\frac{r_{2}(n)}{(n-\tilde{n}_{j})^{2}}\ll
L^{1+o(1)}.$

So, we see that

$c:=\liminf_{n\in{\mathcal{N}}^{\prime}}\frac{\log m_{1}(n)}{\log\log
n}\in[\tfrac{1}{2}\log 2,1].$

As a consequence of Prop. 2.5, we see that

$\log\left\langle\tau_{q}\right\rangle_{n}=(1-2q)\log G+O(\log q)+o(1)$

and from the definition of $G$ we have

$\log\left\langle\tau_{q}\right\rangle_{n}=-2q\log\left\langle\Delta_{j}\right\rangle_{n}=-2q\,\alpha\log\log
n+o(\log\log n).$

Combining these equations we obtain

(5.3) $\log G=\alpha(1+o(1))\frac{2q}{2q-1}\log\log n,$

and we recall the requirement $G/\sqrt{\log n}\to+\infty$. Therefore, we must
have $\alpha\frac{2q}{2q-1}<\frac{1}{2}$ which yields the upper bound
$q<(2-4\alpha)^{-1}$.

Since $N=2\pi G$, we obtain

$\displaystyle\lim_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{h_{q}(\tilde{n}_{j})}{\log
N}=$
$\displaystyle\frac{1}{\alpha}(1-\frac{1}{2q})\lim_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{\log
m_{q}(\tilde{n}_{j})}{\log\log\tilde{n}_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(1-\frac{1}{2q})\log 2$

and

$\displaystyle\liminf_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{h_{1}(\tilde{n}_{j})}{\log
N}=$
$\displaystyle\frac{1}{\alpha}(1-\frac{1}{2q})\liminf_{n\in{\mathcal{N}}^{\prime}\to+\infty}\frac{\log
m_{1}(\tilde{n}_{j})}{\log\log\tilde{n}_{j}}$ $\displaystyle=$
$\displaystyle\frac{c}{\alpha}(1-\frac{1}{2q}).$

We, thus, obtain the following formula for the fractal exponent:

(5.4)
$\begin{split}D_{q}=&\limsup_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{H_{q}(\tilde{n}_{j})}{\log
N}\\\
=&\frac{1}{1-q}\left(\lim_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{h_{q}(\tilde{n}_{j})}{\log
N}-q\liminf_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{h_{1}(\tilde{n}_{j})}{\log
N}\right)\\\ =&\frac{1}{2\alpha}\frac{2cq-\log
2}{q-1}\left(1-\frac{1}{2q}\right)\end{split}$

###  5.2. Proof of Theorem 2.2

We know (cf. [32]) that the new eigenvalues of the operator
$-\Delta_{\varphi}$ are on average close to a nearby old Laplace eigenvalue.
More precisely, there exists a full density subsequence
${\mathcal{N}}_{2}\subset{\mathcal{N}}$ such that $\Delta_{m}=O(1/\sqrt{\log
m})$.

We have

$\sum_{n\in{\mathcal{N}}}\frac{r_{2}(n)}{|n-\lambda_{m}|^{2q}}=\frac{r_{2}(\tilde{n}_{j})}{|\tilde{n}_{j}-\lambda_{j}|^{2q}}+\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\
m\neq\tilde{n}_{j}\end{subarray}}\frac{r_{2}(m)}{|m-\lambda_{j}|^{2q}}$

and, by applying Lemma 4.1, we get for $\tilde{n}_{j}\in{\mathcal{N}}_{1}$ and
$q>1$

$\sum_{\begin{subarray}{c}m\in{\mathcal{N}}\\\
m\neq\tilde{n}_{j}\end{subarray}}\frac{r_{2}(m)}{|m-\lambda_{j}|^{2q}}=O((\log\tilde{n}_{j})^{-q+1/2+o(1)}).$

If $\tilde{n}_{j}\in{\mathcal{N}}_{1}\cap{\mathcal{N}}_{2}$, then, for any
$q>1$,

$m_{q}(\lambda_{j})=(\tilde{n}_{j}-\lambda_{j})^{2q}\sum_{m\in{\mathcal{N}}}\frac{r_{2}(m)}{|m-\lambda_{j}|^{2q}}=r_{2}(\tilde{n}_{j})+O((\log\tilde{n}_{j})^{-2q+1/2+o(1)})$

and thus the general entropy has asymptotics

$h_{q}(\tilde{n}_{j})\sim\log r_{2}(\tilde{n}_{j})$

and

$d_{q}=\lim_{\tilde{n}_{j}\in{\mathcal{N}}_{1}\to+\infty}\frac{h_{q}(\tilde{n}_{j})}{\tfrac{1}{2}\log
2\cdot\log\log\tilde{n}_{j}}=1$

because $r_{2}(\tilde{n}_{j})=(\log\tilde{n}_{j})^{\tfrac{1}{2}\log 2+o(1)}$.

Moreover, for $q=1$, we recall the bound (5.2) to see that

(5.5)
$m_{1}(\lambda_{j})=r_{2}(\tilde{n}_{j})+O(L^{o(1)})\sim(\log\tilde{n}_{j})^{\tfrac{1}{2}\log
2+o(1)}$

where we used $\Delta_{j}^{2}=O((\log\tilde{n}_{j})^{-1})$. It follows that

$d_{1}=\lim_{\tilde{n}_{j}\in{\mathcal{N}}_{1}\to+\infty}\frac{h_{1}(\tilde{n}_{j})}{\tfrac{1}{2}\log
2\cdot\log\log\tilde{n}_{j}}=1.$

Hence, for $q\neq 1$,

$D_{q}=\frac{d_{q}-qd_{1}}{1-q}=1.$

##  6\. Rigid quantizations

In this section we will deal with quantizations which have the property that
the new eigenvalues stay close to the intermediate values of the Laplace
eigenvalues, $\bar{n}_{j}=\tfrac{1}{2}(n_{j}+n_{j+1})$, in the sense that
$\left\langle\Delta_{j}\right\rangle_{x}\asymp\left\langle\delta_{j}\right\rangle_{x}=(\log
x)^{1/2+o(1)}$.

Under the assumption of an analogue of the Sathe-Selberg asymptotic on
$\omega_{1}(n)$, the number of distinct prime factors congruent to $1$ mod 4,
we obtain the following lemma which improves the exponent in Lemma 4.1 if $q$
is sufficiently large.

######  Lemma 6.1.

Let $q>\tfrac{3}{2}$. There exists a subsequence of full density
${\mathcal{N}}_{3}\subset{\mathcal{N}}$ such that for any
$m\in{\mathcal{N}}_{1}$ we have the bound

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\ n\neq
m\end{subarray}}\frac{r_{2}(n)}{|m-n|^{2q}}\ll(\log m)^{-q+\tfrac{1}{2}\log
2+f(q)+o(1)}$

as $m\to+\infty$, where

$f(q)=\frac{1}{2}\log 2\left(\exp\left\\{\frac{\log
2}{q-1/2}\right\\}-1\right).$

###### Proof.

Let $m\in{\mathcal{N}}_{3}(x)$. For convenience denote $L=\log m$. We first of
all claim that there is a full density subsequence of
${\mathcal{N}}_{3}\subset{\mathcal{N}}$ such that for $m\in{\mathcal{N}}_{3}$
(proof to be given in a separate section) the following properties hold. Note
that properties (ii)-(iv) coincide with the respective properties of
${\mathcal{N}}_{1}$, which we restate here for completeness.

  * (i)

if $r\in[L^{\log 2/2+\eta},L^{2}]$, and $r\in[2^{i}L^{\log
2/2+\eta},2^{i+1}L^{\log 2/2+\eta}]$ for some integer $i$, then there are
$\gg(1+2\eta^{\prime})^{i}/((\log r)^{2}\log L)$ elements in ${\mathcal{N}}$
that lie in between $m$ and $n$ if $2^{\omega_{1}(n)}\geq r$

  * (ii)

if $r\geq L^{2}$ there are $\gg_{\epsilon}r^{5/4-\epsilon}/\log r$ elements in
${\mathcal{N}}$ that lie in between $m$ and $n$

  * (iii)

the nearest neighbours of $m$ are at distance at least $L^{1/2-o(1)}$

  * (iv)

for $G\in[2,m^{1-\epsilon}]$ we have the bound

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}}\\\ |m-n|\geq
G\end{subarray}}|m-n|^{-2q}\leq\frac{(\log G)^{2}}{G^{2q-1}L^{1/2-o(1)}}$

  * (v)

for $f(n):=r_{2}(n)/{4\cdot 2^{\omega_{1}(n)}}$ and $Q\geq 1$ there are at
least $Q/((\log Q)^{2}\log L)$ elements in ${\mathcal{N}}$ that lie in between
$m$ and $n$ if $f(n)\geq Q$

We next divide the sum in three parts according to the size of $r_{2}(n)$:

Small $r_{2}(n)$: $r_{2}(n)\leq L^{\log 2/2+\epsilon}$. We have

$\displaystyle\sum_{\begin{subarray}{c}|m-n|\gg L^{1/2-o(1)}\\\ r_{2}(n)\leq
L^{\log 2/2+\epsilon}\end{subarray}}\frac{r_{2}(n)}{(m-n)^{2q}}\ll$
$\displaystyle\sum_{|m-n|\gg L^{1/2-o(1)}}\frac{r_{2}(n)}{(m-n)^{2q}}$
$\displaystyle\ll$ $\displaystyle\frac{L^{\log
2/2+\epsilon}}{L^{q-1/2-o(1)}L^{1/2-o(1)}}=L^{-q+\log 2/2+\epsilon}$

Medium $r_{2}(n)\in[L^{\log 2/2+\eta},L^{2}]$: Set $R_{i}=2^{i}(\log m)^{\log
2/2+\eta}$. By property (6) (cf. [23]) we may set $T=|m-n|$ and we then
obtain, in view of (i),

$\frac{T(\log T)^{2}}{(\log
m)^{1/2-o(1)}}\gg\frac{(1+2\eta^{\prime})^{i}}{(\log R_{i})^{2}\log L},$

where $\eta^{\prime}=\eta/\log 2$, which implies
$T\gg(1+2\eta^{\prime})^{i}L^{1/2-o(1)}$.

Similarly, in view of (v), if $f(n)=r_{2}(n)/(4\cdot
2^{\omega_{1}(n)})\in[2^{j},2^{j+1}]$, then we have

$\frac{T(\log T)^{2}}{(\log m)^{1/2-o(1)}}\gg\frac{2^{j}}{j^{2}\log L}$

which implies $T\gg 2^{j}L^{1/2-o(1)}$.

Combining these two lower bounds we get

$|m-n|\gg G_{i,j}:=2^{j/2}(1+2\eta^{\prime})^{i/2}L^{1/2-o(1)}.$

We define

${\mathcal{N}}_{i,j}:=\\{n\in{\mathcal{N}}\mid
f(n)\in[2^{j},2^{j+1}]\;\text{and}\;2^{\omega_{1}(n)}\in[R_{i},R_{i+1}]\\}.$

Therefore,

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}},\;n\neq m\\\ r_{2}(n)\in[L^{\log
2/2+\eta},L^{2}]\end{subarray}}\frac{r_{2}(n)}{|m-n|^{2q}}=\sum_{i,j\geq
0}\sum_{n\in{\mathcal{N}}_{i,j}}\frac{r_{2}(n)}{|m-n|^{2q}}\ll\sum_{i,j\geq
0}2^{j}\sum_{n\in{\mathcal{N}}_{i,j}}\frac{2^{\omega_{1}(n)}}{|m-n|^{2q}}$

Let us bound the interior sum:

(6.1)
$\begin{split}\sum_{n\in{\mathcal{N}}_{i,j}}\frac{2^{\omega_{1}(n)}}{|m-n|^{2q}}&\ll\sum_{|m-n|\gg
G_{i,j}}\frac{R_{i}}{|m-n|^{2q}}\\\ &\ll\frac{R_{i}(\log
G_{i,j})^{2}}{L^{q-o(1)}(1+2\eta^{\prime})^{i(q-1/2)}2^{j(q-1/2)}}\\\
&=\frac{2^{i}}{(1+2\eta^{\prime})^{i(q-1/2)}2^{j(q-1/2)})}L^{-q+\log
2/2+\eta+o(1)}\end{split}$

And if $q>\max(\tfrac{3}{2},\tfrac{1}{2}+\frac{\log
2}{\log(1+2\eta^{\prime})})$, we may sum over $i$ and $j$ to obtain

$\sum_{\begin{subarray}{c}n\in{\mathcal{N}},\;n\neq m\\\ r_{2}(n)\in[L^{\log
2/2+\eta},L^{2}]\end{subarray}}\frac{r_{2}(n)}{|m-n|^{2q}}\ll_{\epsilon}L^{-q+\log
2/2+\eta+o(1)}.$

So, if $q>\tfrac{3}{2}$, then we need

$q>\tfrac{1}{2}+\frac{\log 2}{\log(1+2\eta/\log 2)}$

which is achieved by

$\eta>\tfrac{1}{2}\log 2\left[\exp\left\\{\frac{\log
2}{q-\tfrac{1}{2}}\right\\}-1\right].$

Large $r_{2}(n)$: We imitate the argument used in the proof of property (8) in
[23].
Take $R_{i}=2^{i}L^{2}/100$. Suppose $r_{2}(n)\in[R_{i},2R_{i}]$. By property
(ii) we have $|m-n|\gg_{\epsilon}R_{i}^{5/4-\epsilon}$. And, thus, we have

$\sum_{r_{2}(n)\in[R_{i},2R_{i}]}\frac{r_{2}(n)}{|m-n|^{2q}}\ll
R_{i}\sum_{|m-n|\gg_{\epsilon}R_{i}^{5/4-\epsilon}}|m-n|^{2q}\ll_{\epsilon}R_{i}(R_{i}^{5/4-\epsilon})^{1-2q}L^{-1/2+o(1)}$

and this is bounded by $2^{i(9/4-5q/2+\epsilon)}L^{4-5q+\epsilon}$.

Summing over $i$ (recall $q>1$, so $9/4-5q/2<0$) we obtain

$\sum_{r_{2}(n)\geq
L^{2}}\frac{r_{2}(n)}{|m-n|^{2q}}=O_{\epsilon}(L^{4-5q+\epsilon})=O(L^{1-2q})$

because $q>1$ implies $4-5q<1-2q$.

∎

###  6.1. Proof that ${\mathcal{N}}_{3}$ is of full density:

  * (i)

We use the same strategy as in the proof of property (7) in [23]. Given that
$n\in{\mathcal{N}}$ satisfies $r=2^{\omega_{1}(n)}\in[(\log
n)^{\tfrac{1}{2}\log 2+\eta},(\log n)^{2}]$ we have $r\in[2^{i}(\log
n)^{\tfrac{1}{2}\log 2+\eta},2^{i+1}(\log n)^{\tfrac{1}{2}\log 2+\eta}]$ for
some integer $i$. We remove
$(1+2\eta^{\prime})^{i}/((\log\log n)(\log r)^{2})$ elements of
${\mathcal{N}}$ to the left and right of $n$ and denote the set of such
removed elements by $R_{n}$. As was shown in the proof of property (7), it is
sufficient to bound the total number of removed elements in all sets $R_{n}$
such that $n\leq 2x$. If we show that this set is of zero density in
${\mathcal{N}}$ (i.e. its cardinality is $o(x/\sqrt{\log x})$), then we have
shown that the set of remaining elements is of full density in
${\mathcal{N}}$. Moreover, for any $m$ in this full density set and any
$n\in{\mathcal{N}}$ such that $r_{2}(n)\geq r$ and $r\in[(\log
n)^{\tfrac{1}{2}\log 2+\eta},(\log n)^{2}]$ with $r\in[2^{i}(\log
n)^{\tfrac{1}{2}\log 2+\eta},2^{i+1}(\log n)^{\tfrac{1}{2}\log 2+\eta}]$, we
have at least $(1+2\eta^{\prime})^{i}/((\log\log n)(\log r)^{2})$ elements of
${\mathcal{N}}$ between $m$ and $n$.

It remains to be shown that the union of $R_{n}$, $n\leq 2x$, is indeed of
cardinality $o(x/\sqrt{\log x})$. As in the proof of (7), it is sufficient (in
view of the upper bound $(\log n)^{2}$) to consider $n\in[x/(\log x)^{10},2x]$
and we, thus, have $\log n=(1+o(1))\log x$.

Recall that $\omega_{1}(n)$ denotes the number of distinct prime factors
congruent to $1$ mod 4. The analogue of the Sathe-Selberg asymptotic for
$\omega_{1}(n)$ gives

(6.2) $\\#\\{n\in{\mathcal{N}}(x)\mid\omega_{1}(n)=k\\}\sim C_{0}\frac{x}{\log
x}\frac{(\log\log x)^{k-1}}{2^{k-1}(k-1)!}$

for $k\leq C\log\log x$, where $C$ is a constant.

Let $r=2^{\omega_{1}(n)}\in[R_{i},R_{i+1}]$ with $R_{i}=2^{i}(\log
x)^{\tfrac{1}{2}\log 2+\eta}$

Because $\omega_{1}(n)=\log r/\log 2$, we see that $r\in[R_{i},R_{i+1}]$ is
equivalent to

$\omega_{1}(n)\in[(\tfrac{1}{2}+\eta^{\prime})\log\log
x+i,(\tfrac{1}{2}+\eta^{\prime})\log\log x+i+1]$

where $\eta^{\prime}=\eta/\log 2$. Thus, the asymptotic (6.2) yields that the
number of $n\in{\mathcal{N}}$ satisfying $r\in[R_{i},R_{i+1}]$ is bounded by

$\ll\left(\frac{1}{1+2\eta^{\prime}}\right)^{i}\frac{x}{\sqrt{\log
x}\sqrt{\log\log x}}\ll\frac{x}{(1+2\eta^{\prime})^{i}\sqrt{\log x}}.$

So the total number of removed elements is

$\ll\frac{x}{(1+2\eta^{\prime})^{i}\sqrt{\log
x}}\frac{(1+2\eta^{\prime})^{i}}{(\log\log x)(\log
R_{i})^{2}}\ll\frac{x}{\sqrt{\log x}(\log\log x)i^{2}}$

and summing over $i$ shows that the total number of elements removed is
$o(x/\sqrt{\log x})$.

  * (ii)-(iv)

cf. proof for ${\mathcal{N}}_{1}$.

  * (v)

First of all we have the bound (cf. [23])

$\sum_{n\in{\mathcal{N}}(x)}f(n)\ll\frac{x}{\sqrt{\log x}}.$

So Chebyshev gives that the number of $n\in{\mathcal{N}}(x)$ s. t. $f(n)\geq
T$ is $\ll x/(T\sqrt{\log x})$.

Let us denote ${\mathcal{N}}_{j}=\\{n\in{\mathcal{N}}\mid
f(n)\in[2^{j},2^{j+1}]\\}$. For each $n\in{\mathcal{N}}_{j}$ let us remove
$2^{j}/(j^{2}\log\log n)$ elements of ${\mathcal{N}}$ to the right and left.

Let us bound the total number of removed elements in ${\mathcal{N}}(x)$ for
the set ${\mathcal{N}}_{j}$. It is sufficient to do this for elements
$n\geq\sqrt{x}$, so that $\log\log n=(1+o(1))\log\log x$. Then, in view of
Chebyshev, the total number of elements removed is

$\ll\frac{x}{2^{j}\sqrt{\log x}}\cdot\frac{2^{j}}{j^{2}\log\log
x}=\frac{x}{j^{2}\sqrt{\log x}\cdot\log\log x}$

and summing over $j$ shows that the total number of elements removed for all
sets ${\mathcal{N}}_{j}$ is

$\ll\frac{x}{\sqrt{\log x}\cdot\log\log x}=o(|{\mathcal{N}}(x)|).$

###  6.2. Breakdown of multifractality

For rigid quantizations which have the property that the new eigenvalues stay
close to the intermediate values of the old Laplace eigenvalues, in the sense
that

$\left\langle\delta_{j}\right\rangle_{x}\asymp\left\langle\Delta_{j}\right\rangle_{x}=(\log
x)^{1/2+o(1)},$

we may see a breakdown of multifractality.

We will use Lemma 6.1 to obtain an estimate of the fractal exponent $d_{q}$ to
illustrate this phenomenon. Let $L=\log\tilde{n}_{j}$. The lemma yields the
following estimate on the moment sum $m_{q}$:

(6.3)
$\begin{split}m_{q}(\tilde{n}_{j})=~{}&r_{2}(\tilde{n}_{j})+\Delta_{j}^{2q}O(L^{(\log
2/2-q+f(q)+o(1))})\\\ =~{}&L^{\log 2/2+o(1)}+O(L^{\log
2/2+f(q)+o(1)}).\end{split}$

Thus, we find

$\limsup_{n_{j}\in{\mathcal{N}}^{\prime}}\frac{\log m_{q}(n_{j})}{\log\log
n_{j}}\in[\tfrac{1}{2}\log 2,\tfrac{1}{2}\log 2+f(q)].$

and (recall $\alpha=\tfrac{1}{2}$)

$\displaystyle
d_{q}=\limsup_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{h_{q}(\tilde{n}_{j})}{\log
N}=$ $\displaystyle
2(1-\frac{1}{2q})\limsup_{\tilde{n}_{j}\in{\mathcal{N}}^{\prime}\to+\infty}\frac{\log
m_{q}(\tilde{n}_{j})}{\log\log\tilde{n}_{j}}$ $\displaystyle=$
$\displaystyle(1-\frac{1}{2q})(1+F(q))\log 2$

where the function $F$ satisfies the inequality

$0\leq F(q)\leq\frac{2\log 2}{2q-1}$

which does not rule out the possibility that $F(q)=(2q-1)^{-1}$ which in turn
would yield $d_{q}=\log 2$. It is striking that even a precise asymptotic such
as an analogue of Sathe-Selberg for integers representable as sums of two
squares does not seem to be sufficient to improve this bound further. We are,
therefore, led to suspect that multifractality may break down for such rigid
quantizations.

##  7\. Multifractality of the ground state

Consider a general rectangular torus
${\mathbb{T}}^{2}={\mathbb{R}}^{2}/2\pi{\mathcal{L}}_{0}$, where
${\mathcal{L}}_{0}={\mathbb{Z}}(a,0)\oplus{\mathbb{Z}}(0,1/a)$ for some $a>0$.
Denote the dual lattice of ${\mathcal{L}}_{0}$ by ${\mathcal{L}}$. We recall
the $L^{2}$ expansion of the eigenfunction (we drop the $1/(4\pi)^{2}$ factor
for convenience)

$G_{\lambda}(x)=\sum_{\xi\in{\mathcal{L}}}\frac{1}{|\xi|^{2}-\lambda}e^{i\left\langle\xi,x\right\rangle}$

Let us consider the (non-normalized) complex moment

$\zeta_{\lambda}(s)=\sum_{\xi\in{\mathbb{Z}}^{2}}\frac{1}{||\xi|^{2}-\lambda|^{s}},\quad\operatorname{Re}s>1$

which gives information about the multifractal properties of the
eigenfunctions.

In order to study the multifractal properties of the ground state of the S̆eba
billiard in the limit $\lambda\to 0$, we first of all note that the leading
term is the coefficient $\lambda^{-s}$ which corresponds to the constant term
(no dependence on $x$) in the $L^{2}$-expansion. We might expect interesting
scaling properties of the fluctuations of the eigenfunction around the
constant term (effectively, we are studying the scaling properties of the
regularized function $G_{\lambda}+1/\lambda$).

Let us, therefore, study the modified moment

$\zeta_{\lambda}^{*}(s)=\sum_{\xi\in{\mathcal{L}}\setminus\\{0\\}}||\xi|^{2}-\lambda|^{-s},\quad\operatorname{Re}s>1.$

Accordingly, we define the $\ell^{2}$-normalized modified moment as

$M_{s/2}^{*}(\lambda)=\frac{\zeta_{\lambda}^{*}(s)}{\zeta_{\lambda}^{*}(2)^{s/2}}.$

The limit of $M_{s/2}^{*}(\lambda)$, as $\lambda\to 0$, exists and we have

$M_{s/2}^{*}(0)=\frac{\zeta_{Q}(s)}{\zeta_{Q}(2)^{s/2}}$

where $\zeta_{Q}$ denotes the Epstein zeta function for the quadratic form
$Q(m,n)=a^{2}m^{2}+a^{-2}n^{2}$:

$\zeta_{Q}(s)=\sum_{(m,n)\neq(0,0)}Q(m,n)^{-s}$

In the half-plane $\operatorname{Re}s>1$, we have

$M_{s/2}^{*}(\lambda)=\zeta_{Q}(s)/\zeta_{Q}(2)^{s/2}+\lambda
F(s)+O(\lambda^{2})\times G(s)$

where $F(s)$ and $G(s)$ can be written explicitly in terms of $\zeta_{Q}$.

One may now continue $M_{s/2}^{*}(\lambda)$ analytically to the complex plane
using the functional equation for $\zeta_{Q}$:

$\zeta_{Q}(s)=\varphi_{Q}(s)\zeta_{Q}(1-s),\quad\varphi_{Q}(s)=\pi^{2s-1}\frac{\Gamma(1-s)}{\Gamma(s)}.$

In order to derive the fractal exponent, we recall $q=s/2$ and we expect a
scaling law of the form

$M_{q}^{*}(\lambda)\sim k^{(1-q)D_{q}^{*}},\quad\lambda\to 0$

where $k$ is the effective length of the summation (which is defined for
$\operatorname{Re}s>1$ and is constant, due to the fact that we are dealing
with the ground state and $\lambda$ is small).

Therefore, we define

$d_{q}^{*}=\log\zeta_{Q}(2q)$

as the fractal exponent with respect to the $\ell^{\infty}$-normalization and,
with respect to the $\ell^{2}$-normalization, we define

(7.1) $D_{q}^{*}=\frac{\log\zeta_{Q}(2q)-q\log\zeta_{Q}(2)}{1-q}.$

where, for simplicity, we drop the constant factor $1/(\log k)$ (a
normalization in some sense).

Note that in the limit as $q\to 1$ we obtain the Shannon entropy associated
with the probability measure $\mu_{\lambda}(\xi)=\zeta_{Q}(2)^{-1}|\xi|^{-4}$:

$\lim_{q\to
1}D_{q}^{*}=\log\zeta_{Q}(2)-2\frac{\zeta_{Q}^{\prime}(2)}{\zeta_{Q}(2)}$

which we may rewrite as

$H_{Sh.}(\mu_{\lambda})=-\sum_{\xi\in{\mathbb{Z}}^{2}}\frac{1}{\zeta_{Q}(2)|\xi|^{4}}\log\left(\frac{1}{\zeta_{Q}(2)|\xi|^{4}}\right).$

Symmetry around $q=\tfrac{1}{4}$: The functional equation for $\zeta_{Q}$
yields a similar property for $d_{q}^{*}$, $D_{q}^{*}$ with respect to the
transformation $q\mapsto\tfrac{1}{2}-q$.

We have

$d_{1/2-q}^{*}=d_{q}^{*}+\log\varphi_{Q}(2q)$

and as an immediate consequence

(7.2)
$\begin{split}D_{1/2-q}^{*}=&\frac{d_{1/2-q}^{*}+\log\varphi_{Q}(2q)-(\tfrac{1}{2}-q)d_{1}^{*}}{\tfrac{1}{2}+q}\\\
=&\frac{1-q}{\tfrac{1}{2}+q}\left(D_{q}^{*}+\frac{\log\varphi_{Q}(2q)+(2q-\tfrac{1}{2})\log\zeta_{Q}(2)}{1-q}\right).\end{split}$

##  8\. Proof of Proposition 2.5

We recall the definition

$\left\langle f\right\rangle_{T}=\frac{1}{T}\int_{T}^{2T}f(t)dt.$

So let us calculate $\left\langle\tau_{q}\right\rangle_{T}$. We define the
summation into two ranges: $|m-t|\leq T$ and its complement in
${\mathcal{N}}$.

Let us denote

$\tau_{q}^{t}(t,G)=\sum_{G\leq|m-t|\leq T}r_{2}(m)|m-t|^{-2q}$

and denote the sum over the complement $\tau_{q}^{r}(t,G)$.

First of all note that the contribution from $\tau_{q}^{r}$ is negligible
(note $|t|\leq T$:

$\displaystyle\sum_{|m-t|>T}r_{2}(m)|m-t|^{-2q}\ll_{\epsilon}\int_{|x-t|>T}\frac{x^{\epsilon}}{|x-t|^{2q}}dx$
$\displaystyle=\int_{|s|>T}\frac{|s+t|^{\epsilon}}{|s|^{2q}}ds=O_{\epsilon}(T^{1-2q+\epsilon}).$

Let us now calculate the mean of $\tau_{q}^{t}$:

(8.1)
$\left\langle\tau_{q}^{t}\right\rangle_{T}=\frac{1}{T}\sum_{m\in[0,3T]}r_{2}(m)\int_{T}^{2T}|t-m|^{-2q}~{}\mathbbm{1}(t\mid
G\leq|t-m|\leq T)dt$

where the summation is restricted to $m\in[0,3T]$, because for $m$ outside
this interval we have $[T,2T]\cap[m-T,m+T]=\emptyset$.

The main contribution will arise from terms $m\in[T,2T]$. We will deal with
these terms first and rewrite the integral inside the sum as

$I_{m}=\int_{T-m}^{2T-m}|\tau|^{-2q}~{}\mathbbm{1}(\tau\mid G\leq|\tau|\leq
T)d\tau$

We may evaluate (recall $m\geq T$)

$I_{m}^{-}=\int_{T-m}^{0}|\tau|^{-2q}~{}\mathbbm{1}(\tau\mid G\leq|\tau|\leq
T)d\tau=\begin{cases}\int_{T-m}^{-G}|\tau|^{-2q}d\tau,\quad\text{if}~{}m\geq
T+G,\\\ \\\ 0,\quad\text{otherwise.}\end{cases}$

which yields

$I_{m}^{-}=\begin{cases}\frac{1}{1-2q}(-G^{1-2q}+(m-T)^{1-2q}),\quad\text{if}~{}m\geq
T+G,\\\ \\\ 0,\quad\text{otherwise.}\end{cases}$

Similarly, we may evaluate

(8.2)
$\begin{split}I_{m}^{+}=&\int^{2T-m}_{0}\tau^{-2q}~{}\mathbbm{1}(\tau\mid
G\leq|\tau|\leq T)d\tau\\\
=&\begin{cases}\frac{1}{1-2q}((2T-m)^{1-2q}-G^{1-2q}),\quad\text{if}~{}m\leq
2T-G,\\\ \\\ 0,\quad\text{otherwise.}\end{cases}\end{split}$

Let us now calculate the sums over $m$.

We have

(8.3)
$\begin{split}\sum_{m\in[T+G,2T]}r_{2}(m)I_{m}^{-}=&~{}\frac{1}{2q-1}G^{1-2q}\sum_{m\in[T+G,2T]}r_{2}(m)\\\
&~{}+\frac{1}{1-2q}\sum_{m\in[T+G,2T]}r_{2}(m)(m-T)^{1-2q}\end{split}$

and

(8.4)
$\begin{split}\sum_{m\in[T,2T-G]}r_{2}(m)I_{m}^{+}=&~{}\frac{1}{2q-1}G^{1-2q}\sum_{m\in[T,2T-G]}r_{2}(m)\\\
&~{}-\frac{1}{2q-1}\sum_{m\in[T,2T-G]}r_{2}(m)(2T-m)^{1-2q}\end{split}$

and we estimate

$\sum_{m\in[T+G,2T]}r_{2}(m)(m-T)^{1-2q}\ll_{\epsilon}\int_{T+G}^{2T}t^{\epsilon}(t-T)^{1-2q}dt=\int_{G}^{T}(t+T)^{\epsilon}t^{1-2q}dt$

which is $O_{\epsilon}((G+T)^{\epsilon}G^{2-2q})$.

Similarly, one may show that

$\sum_{m\in[T,2T-G]}r_{2}(m)(2T-m)^{1-2q}=O_{\epsilon}((G+T)^{\epsilon}G^{2-2q}).$

So we find

$\frac{1}{T}\sum_{m\in[T,2T]}r_{2}(m)I_{m}=\frac{2\pi}{2q-1}G^{1-2q}(1+O(T^{-1+\theta}))$

where $\theta$ denotes the exponent in the circle law:

$\sum_{m\leq T}r_{2}(m)=\pi T+O(T^{\theta})$

The terms $m\in[0,T]\cup[2T,3T]$ give a contribution of lower order.

To see this consider $m\in[0,T]$ (the other case is very similar).

We have

$\int_{T-m}^{2T-m}\tau^{-2q}~{}\mathbbm{1}(\tau\mid G\leq|\tau|\leq T)d\tau$

and since $m\leq T$ implies $[T-m,2T-m]\subset{\mathbb{R}}_{+}$, we can
evaluate this integral as

$=\begin{cases}\int_{T-m}^{T}\tau^{-2q}d\tau,\quad\text{if}\;m\leq T-G,\\\ \\\
\int_{G}^{T}\tau^{-2q}d\tau,\quad\text{if}\;m>T-G.\end{cases}$

Therefore the main contribution is of the form

$\frac{1}{2q-1}G^{1-2q}\frac{1}{T}\sum_{m\in[T-G,T]}r_{2}(m)=\frac{1}{2q-1}G^{1-2q}(\pi
G/T+O(T^{-1+\theta})).$

and we recall $G=o(T)$.

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