# VALENCY OF CERTAIN COMPLEX-VALUED FUNCTIONS

OLUMA ARARSO ALEMU  Department of Mathematics, Addis Ababa University, Addis
Ababa, Ethiopia.
Email<EMAIL_ADDRESS>
###### Abstract

The valence of a function $f$ at a point $z_{0}$ is the number of
distinct,finite solutions to $f(z)=z_{0}.$ In this paper we bound the valence
of complex-valued harmonic polynomials in the plane for some special harmonic
polynomials of the form $f(z)=p(z)\overline{q(z)}$, where $p(z)$ is an
analytic polynomial of degree $n$ and $q(z)$ is an analytic polynomial of
degree $m,$ and $q(z)\neq\alpha p(z)$ for some constant $\alpha.$ Using
techinques of complex dynamics used in the work Sheil-Small and Wilmshurst on
the valence of harmonic polynomial, we prove that the harmonic polynomial
$f(z)=p(z)\overline{q(z)}$ has the valency of $m+n.$

###### keywords:

Analytic polynomials, anti-analytic polynomials, harmonic polynomials, non-
linear elliptic differential equation, logharmonic, Wilmshurst’s conjecture.

##  1 INTRODUCTION

An upper bound on the total number of roots of analytic function can be known
due to the result of the fundamental theorem of algebra, which stipulates that
every non-zero, single-variable, analytic polynomial of degree $n$ with
complex coefficients has

exactly $n$ complex roots, counted with multiplicity.
The argument principle for harmonic functions can be formulated as a direct
generalization of the classical result for analytic functions, see Duren et al
[11]. The winding number of the image curve $f(C)$ about the origin,
$\frac{1}{2\pi}\Delta_{C}~{}argf(z)$ equals the total number of zeros of $f$
in $\mathbb{D},$ counted according to multiplicity where $\mathbb{D}$ is a
plane domain bounded by a rectifiable Jordan curve C, oriented in the
counterclockwise direction

and $f(z)\neq 0$ on C is analytic in $\mathbb{D}$ and continuous in
$\overline{\mathbb{D}}.$
The location and number of the zeros of analytic polynomials as well as
complex-valued harmonic polynomials has been studied by many researchers.
After a basic paper by Clunie and Sheil Small in 1984 [7], the theory of
Harmonic univalent and multivalent functions attracted attention of complex
analysts. Since then the theory has rapidly developed and became an active
branch of complex analysis for researchers. One area of investigation that has
recently become of interest is the number and location of zeros of complex-
valued harmonic polynomials. Clunie and Sheil-Small [7] introduced the family
of complex-valued harmonic functions $f=u+iv$ defined in the unit disk

$\mathbb{D}=\\{z:|z|<1\\},$ where $u$ and $v$ are real harmonic in
$\mathbb{D}.$
Lewy’s Theorem stipulates that if $f$ is a complex valued harmonic function
that is locally univalent in a domain $\mathbb{D}\subset\mathbb{C},$ then its
Jacobian $,J_{f}(z),$ never vanish for all $z\in\mathbb{D}.$ For more
information we can refer to [16]. As an immediate consequence of Lewy’s
Theorem, a complex valued harmonic function $f(z)=h(z)+\overline{g(z)}$ is
locally univalent and sense-preserving if and only if $h^{\prime}(z)\neq 0$
and $|\omega(z)|<1,$ where $\omega(z)$ is a

dilatation function of $f$ defined by
$\omega(z)=\frac{g^{\prime}(z)}{h^{\prime}(z)}.$
Determining the number of zeros of complex-valued harmonic polynomial is an
interesting research area in complex analysis. Suppose $f$ is a polynomial
function of degree $n$ in two variables and let $\mathcal{Z}_{f}$ denote the
number of zeros of $f,$ that is, number of points $z\in\mathbb{C}$ satisfying
$f(z)=0.$ Then for $n>m,$ we have $n\leq\mathcal{Z}_{f}\leq n^{2}.$ The lower
bound is based on the generalized argument principle and is sharp for each $m$
and $n.$ The upper bound follows from applying Bezout’s theorem to the real
and imaginary parts of $f(z)=0$ after noticing that the zeros are isolated,
which was shown by Wilmshurst [17] and this upper bound is sharp in general.
For instance as illustrated by Wilmshurst in the same paper,
$\left(\frac{1}{i}\right)^{n}Q\left(iz+\frac{1}{2}\right)$ is a polynomial
with $n^{2}$ zeros where
$Q(z)=z^{n}+(z-1)^{n}+i\overline{z}^{n}-i(\overline{z}-1)^{n}.$ But it is
natural to ask whether or not it can

be improved for some interesting special classes of polynomials.
Wilmshurst [10] considered the cases of complex-valued harmonic polynomials of
the form $f(z)=p(z)+\overline{q(z)}.$ If $\mathrm{deg}p=n>m=\mathrm{deg}q,$
then as to the question of improving the bound $\mathcal{Z}_{f}\leq n^{2}$
given additional information, Wilmshurst made the conjecture
$\mathcal{Z}_{f}\leq 3n-2+m(m-1).$ This conjecture is stated in [17]. It is
also among the list of open problems in[6]. For $m=n-1$ the upper bound
follows from Wilmshurst’s theorem and examples were also given in [17] showing
that this bound is sharp. For $m=1,$ the upper bound was proved by D.Khavinson
and G.Swiatek [8], and bound was also sharp. For $m=n-3$, the conjectured
bound is $3n-2+m(m-1)=n^{2}-4n+10.$
Using techniques of complex dynamics, it is still possible to improve for some
special harmonic polynomials of the form $f(z)=p(z)\overline{q(z)}$, where
$p(z)$ is an analytic polynomial of degree $n$ and $q(z)$ is an analytic
polynomial of degree $m,$ and $q(z)\neq\alpha p(z)$ for some constant
$\alpha.$ The valency of this type of function is an open problem in [6] and
we solve it here.

######  Definition 1.1.

[1] Let $\mathbb{D}$ be an open unit disk in the complex plane $\mathbb{C}.$
Denote by $\mathcal{H}(\mathbb{D})$ the linear space of all analytic functions
in $\mathbb{D}$, and let $\mathcal{B}(\mathbb{D})$ be the set of all functions
$\omega(z)\in\mathcal{H}(\mathbb{D})$ satisfying
$|\omega(z)|<1,z\in\mathbb{D}.$ A non-constant function $f$ is said to be
logharmonic in $\mathbb{D}$ if $f$ is the solution of the nonlinear elliptic
differential equation

$\overline{f_{\overline{z}}(z)}=\omega(z)\left(\frac{\overline{f(z)}}{f(z)}\right)f_{z}(z)$

where $\omega\in\mathcal{B}(\mathbb{D})$ and the function $\omega(z)$ is
called the second dilatation of $f.$

######  Remark 1.

For $f(z)$ as in the above definition,

  * i.

The Jacobian $,J_{f}$ of $f$ is given by

$J_{f}=|f_{z}|^{2}-|f_{\overline{z}}|^{2}=|f_{z}|(1-|\omega|^{2}),$ (1)

which is positive, and therefore, every non-constant log-harmonic mapping is
sense-preserving and open in $\mathbb{D}.$

  * ii.

If $f$ is a non-vanishing log-harmonic mapping in $\mathbb{D},$ then $f$ can
be expressed as

$f(z)=h(z)\overline{g(z)},$

where $h(z)$ and $g(z)$ are non-vanishing analytic functions in $\mathbb{D}.$

  * iii.

if $f$ vanishes at $z=0$ but is not identically zero, then such a mapping $f$
admits the following representation

$f(z)=z|z|^{2\beta}h(z)\overline{g(z)},$

where $Re\beta>\frac{-1}{2},h,g\in\mathcal{H}(\mathbb{D}),h(0)\neq 0$ and
$g(0)=1$, see [2].

######  Remark 2 (The previous Conjecture).

[10] Let $f(z)=p(z)+\overline{q(z)},$ where $\mathrm{deg}q(z)=m$ and
$\mathrm{deg}p(z)=n,$ satisfy $f(z)\rightarrow\infty$ as $z\rightarrow\infty.$
By conjugating $f(z),$ if necessary, we may assume that $n>m.$ For $m=n$ and
$m=n-1$ it has been shown that the valence bound $n^{2}$ to be sharp, but for
$m<n-1$ it would be surprising if it were still possible to obtain $n^{2}$
valence.The correct bound is $m(m-1)+3n-2$ for $1<m<n-1.$ For $m=1$ this
becomes $\overline{z}+h(z)$ is at most $3n-2$ valent, or equivalently, that
the number of fixed points of the conjugate of an analytic polynomial of
degree $n$ is at most $3n-2.$

######  Theorem 1.2.

[9] If $p(z)$ is a non constant complex polynomial, then $p$ has a zero in the
complex field $\mathbb{C}.$

As an immediate consequence of the Fundamental Theorem of Algebra, if $p(z)$
is a polynomial and $a_{1},a_{2},...,a_{m}$ are its zeros with $a_{j}$ having
multiplicity $k_{j},$ then

$p(z)=c(z-a_{1})^{k_{1}}(z-a_{2})^{k_{2}}...(z-a_{m})^{k_{m}}$

for some constant $c$ and $k_{1}+k_{2}+...+k_{m}$ is the degree of $p.$

######  Theorem 1.3.

[14] Let $f$ and $g$ be relatively prime polynomials in the real variables $x$
and $y$ with real coefficients, and let $\mathrm{deg}h=n$ and
$\mathrm{deg}g=m.$ Then the two algebraic curves $f(x,y)=0$ and $g(x,y)=0$
have at most $mn$ points in common.

A Bezout’s theorem is very useful to bound the number of zeros of complex-
valued harmonic polynomials.

######  Theorem 1.4.

[12] For a harmonic polynomial $f(z)=p(z)+\overline{q(z)}$ with real
coefficients, the equation $f(z)=0$ has at most $n^{2}-n$ solutions that
satisfy $(Rez)(Imz)\neq 0$ where $\mathrm{deg}p=n>m=\mathrm{deg}q.$

If we consider $p(z)=z^{n}+(z-1)^{n}$ and $q(z)=z^{n}-(z-1)^{n},$ then
$f(z)=p(z)+\overline{q(z)}$ has $n^{2}$ number of roots including the root at
$0$ with the multiplicity $n.$ In fact, this is the polynomial that Wilmshurst
used (with a slight perturbation to split the multiple root at the origin) to
show that the maximal bound $n^{2}$ is sharp. This is the reason why in
Theorem 1.4 we only consider roots of the coordinate axes and it yields that
the harmonic polynomial with real coefficients and with the maximal $n^{2}$
number of roots should have at least $n$ roots on the coordinate axes.

######  Theorem 1.5.

[17] Let $f$ be a function harmonic in the (entire) complex plane. If
$liminf_{z\rightarrow\infty}|f|>0,$ then $f$ has finitely many zeros.

######  Theorem 1.6.

[17] If $f(z)=p(z)+\overline{q(z)}$ is a harmonic polynomial such that
$\mathrm{deg}p=n>m=\mathrm{deg}q$ and $\lim_{z\rightarrow\infty}f(z)=\infty,$
then $f(z)$ has at most $n^{2}$ zeros.

The proof of this result can also readily follows from Bezout’s theorem.

The following result is stated and peoved in [13, Theorem 1].

######  Theorem 1.7.

Let $q(z)=z-\overline{p(z)},$ where $p(z)$ is an analytic polynomial with
condition $\mathrm{deg}p=n>1.$ Then $\mathcal{Z}_{q}\leq 3n-2.$

##  2 RESULT

Our main result is the following.

######  Theorem 2.1.

Let $f(z)=p(z)\overline{q(z)}$, where $p(z)$ is an analytic polynomial of
degree $n$ and $q(z)$ is an analytic polynomial of degree $m,$ and let
$q(z)\neq\alpha p(z)$ for some constant $\alpha.$ Then $\mathcal{Z}_{f}\leq
m+n.$

###### Proof.

Since $p(z)$ is an analytic polynomial of degree $n$ and $q(z)$ is an analytic
polynomial of degree $m,$ we have a power series representations:

$p(z)=\sum_{k=0}^{n}a_{k}z^{k}~{}~{}~{}and~{}~{}~{}q(z)=\sum_{j=0}^{m}b_{j}z^{j}.$

Then by the Fundamental Theorem of Algebra, we can decompose $p(z)$ and $q(z)$
as

$p(z)=\beta(z-a_{1})(z-a_{2})...(z-a_{n})~{}~{}~{}and~{}~{}~{}q(z)=\gamma(z-b_{1})(z-b_{2})...(z-b_{m}).$

This directly implies that $p(z)$ has at most $n$ distinct complex roots and
$q(z)$ has at most $m$ distinct complex roots. It then follows from

$f(z)=\underbrace{\underbrace{(\beta\overline{\gamma)}(z-a_{1})(z-a_{2})...(z-a_{n})}_{\leq
n~{}distinct~{}zeros}\underbrace{\overline{(z-b_{1})(z-b_{2})...(z-b_{m})}}_{\leq
m~{}distinct~{}zeros}.}_{\leq n+m~{}distinct~{}zeros.}$

It is not surprising that $q(z)$ and $\overline{q(z)}$ have the same number of
zeros. Therefore, $\mathcal{Z}_{f}\leq m+n.$ ∎

##  3 SHARPNESS OF THE BOUND

######  Example 3.1.

$f(z)=|z|^{2}z+\overline{z}$ is a complex-valued function and can be rewritten
as $f(z)=(z^{2}+1)(\overline{z}).$ In our case, $p(z)=z^{2}+1$ and $q(z)=z.$
We have here, $z=i,z=-i$ and $z=0$ are the three distinct zeros of $f.$
Therefore, the valency $\mathcal{Z}_{f}\leq m+n$ is sharp.

##  4 CONCLUSION

In this paper we have determined the maximum number of the zeros of harmonic
polynomial of the form $f(z)=p(z)\overline{q(z)}$, where $p(z)$ is an analytic
polynomial of degree $n$ and $q(z)$ is an analytic polynomial of degree $m,$
and $q(z)\neq\alpha p(z)$ for some constant $\alpha.$ The result shows that
the valence of $f(z)$ is $m+n.$

## ACKNOWLEDGMENTS

I would like to thank Addis Ababa University, Simons Foundation, and
ISP(International Science Program) at department level for providing us
opportunities and financial support.

## DECLARATION OF INTEREST

The author declare that there are no conflicts of interest regarding the
publication of this paper.

## References

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# Random analytic functions via Gaussian multiplicative chaos

Yichao Huang  Beijing Institute of Technology, School of Mathematics and
Statistics, Beijing, China<EMAIL_ADDRESS>and  Eero Saksman
University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68,
FIN-00014 University of Helsinki, Finland<EMAIL_ADDRESS>The
authors dedicate the paper in honor of Professor Håkan Hedenmalm on his 60th
birthday∗

###### Abstract.

We define a random analytic function $\varphi$ on the unit disc by letting a
Gaussian multiplicative measure to be one of its Clark measures. We show that
$\varphi$ is almost surely a Blaschke product and we provide rather sharp
estimates for the density of its zeroes.

Y.H. is partially supported by National Key R&D Program of China (No.
2022YFA1006300), and is grateful for the support from ERC Advanced Grant
741487 QFPROBA, as most of this work was carried out at the University of
Helsinki. E.S. is supported by the Finnish Academy grant 309940.

∗We have to admit that there is a little delay from our part…

##  1\. Introduction

Given an analytic self map $\varphi:{\mathbb{D}}\to{\mathbb{D}}$ of the unit
disc in the complex plane, for each
$\alpha\in{\mathbb{T}}\coloneqq\partial{\mathbb{D}}$ the measure
$\nu_{\alpha}\coloneqq\nu_{\varphi,\alpha}$ is defined via

(1)
$\mathrm{Re}\left(\frac{\alpha+\varphi(z)}{\alpha-\varphi(z)}\right)\;=\;\int_{{\mathbb{T}}}\frac{1-|z|^{2}}{|e^{i\theta}-z|^{2}}\nu_{\alpha}(d\theta),\qquad
z\in{\mathbb{D}}.$

The measure $\nu_{\alpha}$, or especially its singular part, describes how
strongly and where on the boundary the function $\varphi$ takes the value
$\alpha$ – note that this can happen only at the boundary. The _Clark
measures_ $\nu_{\alpha}$ thus obtained for $\alpha\in{\mathbb{T}}$ have been
much studied especially in connection with applications to perturbative
operator theory, including the spectral theory of $1$-dimensional Schrödinger
equation. Clark measures are defined in [Cla72] and further studied by
Aleksandrov [Ale87, Ale89, Ale94, Ale95, Ale96], so they are also called
Aleksandrov measures. We refer to the reviews [PS06] and [Sak07] for the basic
properties of this interesting family of measures.

In the present paper we will employ the Clark measures to construct random
analytic functions on the unit disc by defining
$\varphi:{\mathbb{D}}\to{\mathbb{D}}$ as the analytic self map of
${\mathbb{D}}$ such that its Clark measure at point $\alpha=1$ equals a given
random measure $\nu$. Especially, we will consider the choice

(2) $\nu_{\varphi,1}=\mu_{\gamma}\coloneqq\text{``}\exp(\gamma
X(\theta))\text{''},\qquad\gamma\in(0,\sqrt{2}].$

Here $\gamma\in(0,\sqrt{2}]$ is a fixed parameter, and $\text{``}\exp(\gamma
X(\theta))\text{''}$ stands for the (random) _Gaussian multiplicative chaos
measure_ corresponding to a log-correlated Gaussian field $X$. The most
natural choice for $X$ on ${\mathbb{T}}$ is the so-called “canonical log-
correlated field”

$X_{c}(\theta)=\sum_{n=1}^{\infty}n^{-1/2}\big{(}A_{n}\cos(n\theta)+B_{n}\sin(n\theta)\big{)},\qquad\theta\in[0,2\pi),$

where the $A_{n},B_{n}$ are independent, identically distributed standard
Gaussians. In general, log-correlated Gaussian fields $X$ on ${\mathbb{T}}$
have the covariance structure

(3)
$\mathbb{E}\left[X(\theta_{1})X(\theta_{2})\right]=\log\frac{1}{|e^{i\theta_{1}}-e^{i\theta_{2}}|}+g(\theta_{1},\theta_{2})$

where $g:{\mathbb{T}}^{2}\to{\mathbb{R}}$ is a (symmetric) continuous
function. For the canonical log-correlated field $X_{c}$, one has $g\equiv 0$.

Multiplicative chaos measures form a natural and important class of random
positive measures that has re-appeared during last $15$ years in various
important roles in statistical physics and other applications. We recall their
proper definition and basic properties in Section 2, and we also recall the
definition of critical chaos measures at the threshold $\gamma=\sqrt{2}$ when
it is needed. Overall, we refer the reader to the review [RV14] for a general
background.

It is known that the mesures $\mu_{\gamma}$ for $\gamma\in(0,\sqrt{2})$ and
$\gamma=\sqrt{2}$ are almost surely purely singular with respect to the
Lebesgue measure on ${\mathbb{T}}$. The basic theory of Clark measures then
implies that $\varphi$ is almost surely an inner function, i.e.
$|\varphi(e^{i\theta})|=1$ for almost every $\theta$, see Proposition 11. Any
inner function $\varphi$ admits the decomposition (see [Gar07, Theorem 5.5])

$\varphi(z)=B(z)S(z),$

where $B$ is a _Blaschke product_ :

$B(z)=e^{ic}z^{m}\prod\limits_{|z_{n}|\neq
0}\frac{\overline{z}_{n}}{|z_{n}|}\frac{z_{n}-z}{1-\overline{z}_{n}z}$

with $c\in{\mathbb{R}}$, $m\geq 0$, $z_{n}\in{\mathbb{D}}$ and
$\sum_{n=1}^{\infty}(1-|z_{n}|)<\infty$, and $S$ is a _singular inner
function_ , i.e.

$S(z)=\exp\left(-\int_{{\mathbb{T}}}\frac{e^{i\theta}+z}{e^{i\theta}-z}d\eta(\theta)\right)$

where the measure $d\eta$ is positive and singular to $d\theta$.

The harmless convention $\mathrm{Im}(\varphi(0))=0$ will be used in this
paper. Since we focus on the case $\alpha=1$, by the Herglotz representation
formula,

(4) $\varphi(z)=\frac{h(z)-1}{h(z)+1},\qquad
h(z)\coloneqq\int_{{\mathbb{T}}}\frac{e^{i\theta}+z}{e^{i\theta}-z}\nu_{1}(d\theta).$

Our first result shows that the singular part of our random function $\varphi$
is almost surely trivial:

######  Theorem 1.

Let $\gamma\in(0,\sqrt{2}]$ (where $\gamma=\sqrt{2}$ refers to the critical
chaos) and let $\mu_{\gamma}$ be the chaos measure on ${\mathbb{T}}$
corresponding to a log-correlated field $X$ with the covariance structure (3).
Assume that $g\in W^{s,2}({\mathbb{T}}^{2})$ for some $s>1$. Then, the inner
function $\varphi$ defined via (4) with the choice $\nu_{1}=\mu_{\gamma}$ is
almost surely a Blaschke product.

This result can be thought of as a random analogue of a famous result of
Frostman [Gar07, Theorem 6.4], which states that given an inner function $f$,
for almost every $a\in{\mathbb{D}}$ the “Frostman shift” $\tau_{a}\circ f$ is
a pure Blaschke product. Here $\tau_{a}$ is the Möbius automorphism
$\tau_{a}:{\mathbb{D}}\to{\mathbb{D}}$, and
$\tau_{a}(z)=\frac{a-z}{1-\overline{a}z}.$

We also provide an estimate for the density of the zeroes of the random
Blaschke product $\varphi$:

######  Theorem 2.

Assume that $g\in W^{2,2}({\mathbb{T}}^{2})$ in (3) and let
$z_{1},z_{2},\dots$ stand for the zeroes of $\varphi$. Then we have the
following phase transition for the density of zeroes:

  1. (1)

Almost surely, $\sum_{k=1}^{\infty}(1-|z_{k}|)^{\beta}<\infty$ for
$1-\gamma^{2}/8<\beta$.

  2. (2)

Almost surely, $\sum_{k=1}^{\infty}(1-|z_{k}|)^{\beta}=\infty$ for
$1-\gamma^{2}/8>\beta$.

The above result is slightly surprising in the sense that the density of the
zeroes goes up when the parameter $\gamma$ of the chaos measure decreases. On
the other hand, it is natural when one notes that in a heuristic way the
support of $\mu_{\gamma}$ “decreases” in a sense when $\gamma$ increases. One
may ask what happens in the supercritical case, especially how does the
analogue of Theorem 2 look like for $\gamma>\sqrt{2}$. One of the questions
that we plan to study in the future is how the zero set behaves realization
wise as the parameter $\gamma$ varies.

The present note is a part of our long-term project to study some aspects of
random spectral theory via Clark measures. We thank Alexei Poltoratski for
reawakening our interest in these questions, and especially we are grateful to
discussions with Håkan Hedenmalm who independently suggested us (during a
visit to the University of Helsinki a couple of years ago) that one should
expect a result to the direction of Theorem 1 above.

##  2\. Log-correlated Gaussian fields and Gaussian multiplicative chaos

In the following, we consider several random (generalized) functions defined
on the unit disc $\mathbb{D}\subset\mathbb{R}^{2}$, on the unit circle
${\mathbb{T}}=\partial\mathbb{D}$, or on an interval of the real axis, say
$\left[-\frac{1}{2},\frac{1}{2}\right]\subset\mathbb{R}$. We will usully
denote by $X,Y,Z,\dots$ some log-correlated Gaussian fields. The parameter
$\gamma\in\left(0,\sqrt{2}\right)$ will be fixed and
$\mu\coloneqq\mu_{\gamma}$ will denote the (subcritical) Gaussian
multiplicative chaos measure with parameter $\gamma$. The critical case
$\gamma=\sqrt{2}$ will be recalled separately in the proofs.

Since this note is intended for readers from both probability and analysis
backgrounds, we recall some definitions and basic facts about log-correlated
Gaussian fields and Gaussian multiplicative chaos.

###  2.1. Log-correlated Gaussian fields on the unit circle

Consider the Gaussian field $X_{c}$ on the unit circle ${\mathbb{T}}$ defined
by a random Fourier series:

(5)
$X_{c}(\theta)=\sum\limits_{n=1}^{\infty}n^{-\frac{1}{2}}(A_{n}\cos(n\theta)+B_{n}\sin(n\theta)),\qquad\theta\in[0,2\pi),$

where $A_{n},B_{n}$ and i.i.d. standard Gaussian variables. We refer to this
particular log-correlated field as the _canonical field_ (on ${\mathbb{T}}$).
Let us recall why the Gaussian field $X_{c}(\theta)$ is log-correlated –
actually its covariance kernel writes in the simple form

(6)
$C(\theta,\theta^{\prime})\coloneqq\mathbb{E}\left[X(\theta)X(\theta^{\prime})\right]=-\log\left|e^{i\theta}-e^{i\theta^{\prime}}\right|.$

Indeed, a direct calculation yields

$C(\theta,\theta^{\prime})=\sum\limits_{n=1}^{\infty}\frac{1}{n}\cos(n(\theta-\theta^{\prime})).$

To check that this is the same as $-\log|e^{i\theta}-e^{i\theta^{\prime}}|$,
write

$-\log\left|e^{i\theta}-e^{i\theta^{\prime}}\right|=-\log\left|1-e^{i(\theta-\theta^{\prime})}\right|=-\frac{1}{2}\log\left(1-e^{i(\theta-\theta^{\prime})}\right)-\frac{1}{2}\log\left(1-e^{-i(\theta-\theta^{\prime})}\right)$

and develop the Taylor series.

Due to the divergence of $-\log\left|e^{i\theta}-e^{i\theta^{\prime}}\right|$
on the diagonal $\theta=\theta^{\prime}$, the Gaussian field $X_{c}$ cannot be
defined as a random function, but only as a random distribution in the sense
of Schwartz. Indeed, one verifies that the field $X_{c}$ almost surely lives
in the negative order Sobolev space $W^{-s,2}$ for all $s>0$. Recall that the
Sobolev spaces $W^{s,2}$ for $s\in{\mathbb{R}}$ are defined as

$W^{s,2}({\mathbb{T}})\coloneqq\left\\{f\in
L^{2}({\mathbb{T}})\leavevmode\nobreak\ ;\leavevmode\nobreak\
||f||_{s,2}^{2}\coloneqq\sum\limits_{n=-\infty}^{\infty}\left(1+|n|^{2}\right)^{s}|\hat{f}(n)|^{2}<\infty\right\\}.$

Then $W^{-s,2}$ and $W^{s,2}$ are mutual duals under the natual pairing. For
relevant properties of these classical function spaces $W^{s,2}$ with
$s\in\mathbb{R}$ in relation to Gaussian multiplicative chaos, we refer to
[JSW19, Section 2.2].

We will consider more general log-correlated covariance kernels of the form

$K(\theta,\theta^{\prime})=C(\theta,\theta^{\prime})+g(\theta,\theta^{\prime})$

with $g\in W^{s,2}({\mathbb{T}}^{2})$ for some $s>1$. The Gaussian field $X$
on ${\mathbb{T}}$ with kernel $K$ is also called a log-correlated Gaussian
field, and the above regularity remark applies also to $X$.

###  2.2. Gaussian multiplicative chaos on the unit circle

Fix $\gamma\in(0,\sqrt{2})$. The theory of Gaussian multiplicative measures
developed by Kahane [Kah85] as well as its recent developments take care of
rigorously exponentiating a log-correlated Gaussian field such as $\gamma X$.

######  Definition 3 (Gaussian multiplicative chaos measures).

Let $Y$ denote any log-correlated Gaussian field on ${\mathbb{T}}$ with
covariance kernel

$K_{Y}(e^{i\theta},e^{i\theta^{\prime}})=-\log\left|e^{i\theta}-e^{i\theta^{\prime}}\right|+g\left(e^{i\theta},e^{i\theta^{\prime}}\right)$

where $g$ is a (symmetric) continuous function on
${\mathbb{T}}\times{\mathbb{T}}$.

Let $Y_{\varepsilon}$ be a standard $\varepsilon$-mollification of $Y$ using a
compact and smooth test function. The Gaussian multiplicative chaos measure
$\mu_{Y}$ on ${\mathbb{T}}$, associated to $Y$ with parameter
$\gamma\in\left(0,\sqrt{2}\right)$, is defined as the following limit taken in
probability:

$d\mu_{Y}(\theta)\coloneqq\lim\limits_{\varepsilon\to 0}e^{\gamma
Y_{\varepsilon}(e^{i\theta})-\frac{\gamma^{2}}{2}\mathbb{E}\left[Y_{\varepsilon}(e^{i\theta})^{2}\right]}d\theta.$

Here the convergence of the measures is in the sense of weak∗ convergence,
i.e. for any continuous test function $\phi:{\mathbb{T}}\to\mathbb{R}$ one has

$\int_{{\mathbb{T}}}\phi(e^{i\theta})d\mu_{Y}(\theta)=\lim\limits_{\varepsilon\to
0}\int_{{\mathbb{T}}}e^{\gamma
Y_{\varepsilon}(e^{i\theta})-\frac{\gamma^{2}}{2}\mathbb{E}\left[Y_{\varepsilon}(e^{i\theta})^{2}\right]}\phi(e^{i\theta})d\theta.$

The Gaussian multiplicative chaos measure $\mu_{Y}$ is unique (i.e. does not
depend on the mollification) and non-trivial for all $\gamma^{2}<2$ (we refer
to this as the subcritical regime) and degenerate (i.e. almost surely zero
everywhere) if $\gamma^{2}\geq 2$.

The convergence in law or in probability for Gaussian multiplicative chaos
measures (i.e. that the above definition makes sense) was established
essentially in [Kah85], and later on the theory has been developed in many
works, e.g. [BM03, RV10, Sha16, Ber17].

######  Remark 4 ($1d$-Gaussian multiplicative chaos measures).

The more classical setting, completely similar to the above proposition,
defines the Gaussian multiplicative chaos measures on a compact subset of the
real line $\mathbb{R}$. In this note, we will use mostly the interval
$\left[-\frac{1}{2},\frac{1}{2}\right]\in\mathbb{R}$. Both of them are
$1d$-Gaussian multiplicative chaos measures, as are the chaos measures defined
on ${\mathbb{T}}$.

We first record some important known properties of the Gaussian multiplicative
chaos measures in one-dimension, and we will provide references after stating
the results.

######  Fact 5 (Existence of moments).

Let $Y$ be a log-correlated Gaussian field on ${\mathbb{T}}$ or
$\left[-\frac{1}{2},\frac{1}{2}\right]$ and $\mu_{Y}$ the associated Gaussian
multiplicative chaos measure with $\gamma\in\left(0,\sqrt{2}\right)$ as in
Definition 3. Then for all sets $T\subset{\mathbb{T}}$ or
$T\subset\left[-\frac{1}{2},\frac{1}{2}\right]$ with non-empty interior,

$\mathbb{E}\left[\mu_{Y}(T)^{p}\right]<\infty$

if and only if $p<\frac{2}{\gamma^{2}}$ (this includes finiteness of all
negative moments).

Especially one may note that in the subcritical regime $\gamma^{2}<2$, the
first moment of the mass of a $1d$-Gaussian multiplicative chaos measure
always exists.

######  Fact 6 (Exact scaling log-correlated fields).

Consider the log-correlated Gaussian field $Z$ on the interval
$\left[-\frac{1}{2},\frac{1}{2}\right]$ with covariance kernel:

$K_{Z}(\theta,\theta^{\prime})=\log\frac{1}{|\theta-\theta^{\prime}|}.$

Then $Z$ is (locally) translation invariant and exact scale invariant: in
particular, for all $0<r<1$, the following Gaussian fields are equal in law:

$\left(Z(\theta)+\sqrt{-\log
r}N\right)_{\theta\in\left[-\frac{1}{2},\frac{1}{2}\right]}\qquad\text{and}\qquad\left(Z\left(r^{-1}\theta\right)\right)_{\theta\in\left[-\frac{r}{2},\frac{r}{2}\right]}.$

where $N$ is a standard Gaussian independent of the field $Z$.

As a consequence, the following equality in law holds for the Gaussian
multiplicative chaos measure $\mu_{Z}$ associated to the field $Z$ with
parameter $\gamma\in\left(0,\sqrt{2}\right)$ and all
$[x,x+r]\subset\left[-\frac{1}{2},\frac{1}{2}\right]$:

$\int_{[x,x+r]}d\mu_{Z}(\theta)\sim r^{1+\gamma^{2}/2}e^{\gamma\sqrt{-\log
r}N}\int_{\left[-\frac{1}{2},\frac{1}{2}\right]}d\mu_{Z}(\theta).$

We refer to the last identity as “exact scaling” relation in the sequel.

######  Fact 7 (Kahane’s convexity inequality).

Let $Y_{1}$, $Y_{2}$ be two continuous centered Gaussian processes indexed by
${\mathbb{T}}$ or on any interval $I\subset{\mathbb{R}}$, such that for all
$\theta,\theta^{\prime}$,

$\mathbb{E}\left[Y_{1}(\theta)Y_{1}(\theta^{\prime})\right]\leq\mathbb{E}\left[Y_{2}(\theta)Y_{2}(\theta^{\prime})\right].$

Then, for all convex function $F$ with at most polynomial growth at infinity
and any positive finite measure $\sigma$ on ${\mathbb{T}}$ or on $I$,

$\mathbb{E}\left[F\left(\int
e^{Y_{1}(\theta)-\mathbb{E}[Y_{1}(\theta)^{2}]}d\sigma(\theta)\right)\right]\leq\mathbb{E}\left[F\left(\int
e^{Y_{2}(\theta)-\mathbb{E}[Y_{2}(\theta)^{2}]}d\sigma(\theta)\right)\right].$

###### Proofs of the above properties.

See respectively [RV14, Theorem 2.5], [RV14, Theorem 2.11], [RV14, Theorem
2.16] and [RV14, Theorem 2.1]. To apply the last convexity inequality to log-
correlated Gaussian fields, we systematically use the regularization procedure
in Proposition 3 to meet the continuity assumption: this is standard and we
will not write out the regularization everytime. ∎

######  Remark 8 (Comparison of moments).

For people with background in analysis we now demonstrate a standard use of
Kahane’s convexity inequality mainly for comparing moments of two different
chaos measures, since we need to do this later on with careful book-keeping of
the constants involved. Consider thus $p<\frac{2}{\gamma^{2}}$ and the
function $F(x)=x^{p}$ for $x>0$. Then either $F$ or $-F$ is convex. Consider a
general log-correlated Gaussian field $Y$ as in Definition 3 and an exact
scaling log-correlated Gaussian field $Z$ as in Fact 6. Since the correction
term $g(\theta,\theta^{\prime})$ in the covariance function of the field $Y$
is bounded in absolute value by a finite constant $C$, we can apply Kahane’s
inequality to the following Gaussian fields (with $N$ a normal Gaussian
variable independent of $Y$ and $Z$):

$\left(Y(\theta)+\sqrt{C}N\right)_{\theta}\qquad\text{and}\qquad\left(Z(\theta)\right)_{\theta},$

or

$Y(\theta)\qquad\text{and}\qquad\left(Z(\theta)+\sqrt{C}N\right)_{\theta},$

to conclude that there exists some positive constant $K$ such that, for all
positive functions $P\left(e^{i\theta}\right)$,

$\frac{1}{K}\mathbb{E}\left[\left(\int_{{\mathbb{T}}}P(e^{i\theta})d\mu_{Y}(\theta)\right)^{p}\right]\leq\mathbb{E}\left[\left(\int_{{\mathbb{T}}}P(e^{i\theta})d\mu_{Z}(\theta)\right)^{p}\right]\leq
K\mathbb{E}\left[\left(\int_{{\mathbb{T}}}P(e^{i\theta})d\mu_{Y}(\theta)\right)^{p}\right].$

This relation tells us that modulo a multiplicative constant, the moments of
the positively weighted mass of a general Gaussian multiplicative chaos
measure associated to any log-correlated Gaussian field $Y$ scales similarly
like in the exact scaling case with the field $Z$. We will write the above
equation as

$\mathbb{E}\left[\left(\int_{{\mathbb{T}}}P(e^{i\theta})d\mu_{Z}(\theta)\right)^{p}\right]\asymp\mathbb{E}\left[\left(\int_{{\mathbb{T}}}P(e^{i\theta})d\mu_{Y}(\theta)\right)^{p}\right].$

The constant in $\asymp$ only depends on the parameter $\gamma$, the
correction term $g\left(e^{i\theta},e^{i\theta^{\prime}}\right)$, and $p$.

To summarize, when estimating the moment of a positively weighted mass of a
Gaussian multiplicative chaos measure $\mu_{Y}$, we can switch the underlying
Gaussian field to an exact scaling log-correlated Gaussian field $Z$ of Fact
6, up to a deterministic multiplicative constant.

When $\mu=\mu_{X_{c}}$ is the Gaussian multiplicative chaos measure on
${\mathbb{T}}$ associated to $X$ with the canonical covariance kernel in (6),
the measure $d\mu(\theta)$ is then invariant under rotations of the unit
circle. In fact, with this particular choice, the measure $\mu$ transforms
nicely under any Möbius transformation of the unit disc $\mathbb{D}$. Since we
don’t need this fact as we study general log-correlated Gaussian fields, we
leave the interested reader to consult [Var17] for the fascinating connections
between Gaussian multiplicative chaos measures and Conformal Field Theory.

We record a simple auxiliary result.

######  Lemma 9 (Negative moments).

Let $I_{1},I_{2}$ be two open intervals of the unit circle ${\mathbb{T}}$ or
the interval $\left[-\frac{1}{2},\frac{1}{2}\right]$ and let $\mu$ be any
Gaussian multiplicative chaos measure on ${\mathbb{T}}$ or
$\left[-\frac{1}{2},\frac{1}{2}\right]$ with parameter
$\gamma\in\left(0,\sqrt{2}\right)$. Then the almost surely positive random
variable

$\inf\\{\mu(I_{1}),\mu(I_{2})\\}$

has negative moments of any order.

###### Proof.

For $\gamma\in(0,\sqrt{2}]$, any $1d$-Gaussian multiplicative chaos measure
$\mu$ has all orders of negative moments on any open interval by Fact 5. The
lemma follows by using

$\left(\inf\\{a,b\\}\right)^{-p}\leq a^{-p}+b^{-p}$

for all $a,b>0$ and taking expectations. ∎

######  Lemma 10 (Structure of the carrier).

For $\gamma\in\left(0,\sqrt{2}\right)$, any Gaussian multiplicative chaos
measure on ${\mathbb{T}}$ is almost surely singular with respect to the
Lebesgue measure and does not possess atoms.

###### Proof.

See [RV14, Theorem 2.6] for a proof and furthur discussions. ∎

It follows immediately that the analytic self-map
$\varphi:{\mathbb{D}}\to{\mathbb{D}}$ (defined via (2) and (1)) is a.s. an
inner function, as alluded in the introduction.

######  Proposition 11 (Inner function).

The random analytic function $\varphi$ is almost surely an inner function,
i.e. for all $z\in{\mathbb{T}}$, we have $|\varphi(z)|=1$.

###### Proof.

It is a classical fact [Sak07, Theorem 2.2] on Clark measures that $\varphi$
is an inner function if and only if the Clark measure $\nu_{\varphi,1}$ is
singular with respect to the Lebesgue measure. The claim follows from Lemma
10. ∎

##  3\. Blaschke product property for the canonical field

We find it instrumental to first provide a short and simple proof of the
Blaschke product property (see Theorem 1) in the case of the canonical log-
correlated Gaussian field $X_{c}$. In addition, it prepares the reader gently
for the perturbation philosophy applied also in the more involved proofs in
the sequel.

###  3.1. A Frostman type lemma

To prove the Blaschke product property, we first provide a variant of the
classical Frostman lemma [Gar07, Theorem II.6.4] adapted to our situation.

######  Lemma 12 (A Frostman type lemma).

Let $\varphi$ be a random inner function111Naturally we assume that the map
$(z,\omega)\mapsto\varphi(z)$ is jointly measurable, where $\omega$ stands for
an element in the probability space. on the unit disc. Then $\varphi$ is
almost surely a Blaschke product if for some $\varepsilon>0$

(7)
$\sup\limits_{z\in\mathbb{D}}\mathbb{E}\left(\log\frac{1}{|\varphi(z)|}\right)^{1+\varepsilon}<\infty.$

###### Proof.

The inner function $\varphi$ is a Blaschke product if and only if (see [Gar07,
Theorem II.2.4, Corollary I.6.6])

$\liminf\limits_{r\to
1^{-}}\int_{0}^{2\pi}\log\left(\frac{1}{|\varphi(re^{i\theta})|}\right)d\theta=0.$

Define $F:{\mathbb{D}}\to[0,\infty]$ by setting
$F(z)\coloneqq\mathbb{E}\log\left(\frac{1}{|\varphi(z)|}\right)$. Since almost
surely $\varphi$ is an inner function, and hence a.s. its boundary values are
unimodular a.e. at the boundary $\partial{\mathbb{D}}$, we deduce by Fubini
that for a.e. $\theta\in[0,2\pi]$ it holds that a.s. $\lim_{r\to
1}\log(1/|\varphi(re^{i\theta})|)=0$. By our assumption, we may apply uniform
integrability for the expectation in the definition of $F(re^{i\theta})$ and
deduce that

$\lim_{r\to 1^{-}}F(re^{i\theta})=0\qquad\textrm{for almost
every}\qquad\theta\in[0,2\pi).$

Thereafter (note that $F$ is bounded under our assumption) we may apply
dominated convergence theorem to deduce that $\lim_{r\to
1^{-}}\int_{0}^{2\pi}F(re^{i\theta})d\theta=0$, whereafter Fubini’s theorem
and Fatou’s lemma yield that

$\mathbb{E}\left[\liminf\limits_{r\to
1^{-}}\int_{0}^{2\pi}\log\left(\frac{1}{|\varphi(re^{i\theta})|}\right)d\theta\right]\leq\liminf\limits_{r\to
1^{-}}\int_{0}^{2\pi}F(re^{i\theta})d\theta=0.$

This finishes the proof. ∎

Since $\mathrm{Re}(h(z))\geq 0$, the quantity inside the expectation only
blows up when $h(z)$ is close to $1$. It is then an elementary exercise to
verify that the following condition suffices:

######  Corollary 13.

Let $\varphi$ be defined via Equation (4) with a random singular measure
$\mu$. Then $\varphi$ is almost surely a Blaschke product if for some
$\varepsilon>0$

$\sup\limits_{z\in\mathbb{D}}\mathbb{E}\left(\log^{+}\frac{1}{|h(z)-1|}\right)^{1+\varepsilon}<\infty.$

######  Remark 14.

Since in the proof of Lemma 12 we used the assumption only to guarantee
uniform integrability under the expectation, and at the same time boundedness
of $F$, we could equally well replace the condition (7) by

$\sup\limits_{z\in\mathbb{D}}\mathbb{E}\;\psi\left(\log\frac{1}{|\varphi(z)|}\right)<\infty,$

where $\psi:[0,\infty)\to[0,\infty)$ is any increasing function with
$\lim_{x\to\infty}\psi(x)/x=\infty.$

###  3.2. Proof of the Blaschke product property

We now prove Theorem 1 for the canonical log-correlated Gaussian field
$X_{c}$. Since the measure $\mu$ is invariant under rotations of
${\mathbb{T}}$, it is enough to show that

$\sup\limits_{r\in[0,1)}\mathbb{E}\left(\log^{+}\frac{1}{|h(r)-1|}\right)^{1+\varepsilon}<\infty.$

Since for any $p\in(0,1)$ and $\varepsilon>0$ we have

$\left(\log^{+}\frac{1}{x}\right)^{1+\varepsilon}\leq\frac{C(p,\varepsilon)}{x^{p}}\qquad\text{and}\qquad|h(r)-1|\geq|\mathrm{Im}(h(r))|,$

it is sufficient to consider the imaginary part $\mathrm{Im}(h(r))$ and show
that for some $p\in(0,1)$,

$\sup\limits_{r\in[0,1)}\mathbb{E}\left[\frac{1}{|\mathrm{Im}(h(r))|^{p}}\right]<\infty.$

Notice that the imaginary part $\mathrm{Im}(h(r))$ writes as a signed Gaussian
multiplicative chaos measure:

$\mathrm{Im}(h(r))=\int_{{\mathbb{T}}}\frac{-2r\sin(\theta)}{|r-e^{i\theta}|^{2}}d\mu(\theta).$

The key idea is to take out a Fourier mode in the canonical field $X_{c}$ and
write the following independent decomposition of $X_{c}$:

$X_{c}(\theta)=B_{1}\sin(\theta)+\widetilde{X}(\theta).$

It follows that (where $\widetilde{\mu}$ is the Gaussian multiplicative
measure associated with the log-correlated Gaussian field $\widetilde{X}$):

$d\mu(\theta)=\exp\left(\gamma
B_{1}\sin(\theta)-\frac{\gamma^{2}\sin^{2}(\theta)}{2}\right)d\widetilde{\mu}(\theta),$

and the following derivative has constant sign:

$-\frac{\partial\mathrm{Im}(h(r))}{\partial
B_{1}}=\int_{{\mathbb{T}}}\frac{2\gamma
r\cdot\sin^{2}(\theta)}{|r-e^{i\theta}|^{2}}\cdot\exp\left(\gamma
B_{1}\sin(\theta)-\frac{\gamma^{2}\sin^{2}(\theta)}{2}\right)d\widetilde{\mu}(\theta)\geq
0.$

Now if $B_{1}\geq 0$, by looking at the interval where $\sin(\theta)\geq 0$,
the integral is bigger than

$e^{-\gamma^{2}/2}\int_{\theta\in[0,\pi]}\frac{2\gamma
r\cdot\sin^{2}(\theta)}{|r-e^{i\theta}|^{2}}d\widetilde{\mu}(\theta).$

Similarly if $B_{1}\leq 0$, by looking at the interval where $\sin(\theta)\leq
0$, the integral is bigger than

$e^{-\gamma^{2}/2}\int_{\theta\in[\pi,2\pi]}\frac{2\gamma
r\cdot\sin^{2}(\theta)}{|r-e^{i\theta}|^{2}}d\widetilde{\mu}(\theta).$

In any case, for all $B_{1}\in\mathbb{R}$ and $r\in[1/2,1)$,

$-\frac{\partial\mathrm{Im}(h(r))}{\partial B_{1}}\geq
C\inf\left\\{\int_{\theta\in[\frac{\pi}{4},\frac{3\pi}{4}]}d\widetilde{\mu}(\theta),\int_{\theta\in[\frac{5\pi}{4},\frac{7\pi}{4}]}d\widetilde{\mu}(\theta)\right\\}.$

The above is a lower bound on the derivative of $\mathrm{Im}(h(r))$ with
respect to the Gaussian variable $B_{1}$ using the auxiliary field
$\widetilde{X}$. We now record an elementary observation.

######  Lemma 15.

Let $\frac{1}{\sqrt{2\pi}}e^{-\frac{\rho^{2}}{2}}d\rho$ denote the
distribution density of a standard Gaussian random variable. Let
$v:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that
$v^{\prime}(\rho)\geq b>0$ for all $\rho\in\mathbb{R}$. Then

$\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-\frac{\rho^{2}}{2}}\frac{1}{|v(\rho)|^{p}}d\rho\leq
Cb^{-p},$

with $C=C(p)$, for any $p\in(0,1)$.

###### Proof.

Let $x_{0}\in\mathbb{R}$ be the only point where $v(x_{0})=0$. Then

$|v(\rho)-v(x_{0})|\geq b|\rho-x_{0}|$

and

$\int_{\mathbb{R}}e^{-\frac{\rho^{2}}{2}}\frac{1}{|v(\rho)-v(x_{0})|^{p}}d\rho\leq
b^{-p}\left(\int_{\mathbb{R}}e^{-\frac{\rho^{2}}{2}}\frac{1}{|\rho-
x_{0}|^{p}}d\rho\right).$

It remains to see that

$\int_{\mathbb{R}}e^{-\frac{\rho^{2}}{2}}\frac{1}{|\rho-
x_{0}|^{p}}d\rho\leq\int_{x_{0}-1}^{x_{0}+1}\frac{1}{|\rho-
x_{0}|^{p}}d\rho+\int_{-\infty}^{x_{0}-1}e^{-\frac{\rho^{2}}{2}}d\rho+\int_{x_{0}+1}^{\infty}e^{-\frac{\rho^{2}}{2}}d\rho\eqqcolon
C(p)$

which is finite as long as $p\in(0,1)$. ∎

To conclude, let $\rho$ parametrize the Gaussian variable $B_{1}$ and let $b$
be the above lower bound

$b=C\inf\left\\{\int_{\theta\in[\frac{\pi}{4},\frac{3\pi}{4}]}d\widetilde{\mu}(\theta),\int_{\theta\in[\frac{5\pi}{4},\frac{7\pi}{4}]}d\widetilde{\mu}(\theta)\right\\}$

independent of $B_{1}$, and apply Lemma 15. Taking the expectation with
respect to $B_{1}$ is equivalent to integrating over $\rho$, and we get the
following for the conditional expectation on $B_{1}$ with $p\in(0,1)$:

$\mathbb{E}\left[|\mathrm{Im}(h(r))|^{-p}\,\big{|}\,\widetilde{X}\right]\leq
C\left(\inf\left\\{\int_{\theta\in[\frac{\pi}{4},\frac{3\pi}{4}]}d\widetilde{\mu}(\theta),\int_{\theta\in[\frac{5\pi}{4},\frac{7\pi}{4}]}d\widetilde{\mu}(\theta)\right\\}\right)^{-p}.$

Now taking the expectation with respect to all other Gaussian variables,

$\mathbb{E}\left[\frac{1}{|\mathrm{Im}(h(r))|^{p}}\right]\leq
C\mathbb{E}\left[\left(\inf\left\\{\int_{\theta\in[\frac{\pi}{4},\frac{3\pi}{4}]}d\widetilde{\mu}(\theta),\int_{\theta\in[\frac{5\pi}{4},\frac{7\pi}{4}]}d\widetilde{\mu}(\theta)\right\\}\right)^{-p}\right].$

The last quantity, independent of $r$, is finite for all $p\in(0,1)$ by Lemma
9: this finishes the proof.

##  4\. On the density of zeroes: preparations

In this section, we prepare our study for the density problem, Theorem 2.
Section 4.1 first develops a quantitative version of the Frostman-type Lemma
12, and provides a sufficient condition to Theorem 2. It turns out that we
again need uniform estimate for the expectation of a suitable functional with
a log-singularity, acting on the real part of the Poisson extension of the
measure.

In Section 4.2 we then get rid of the log-singularity using a perturbative
method, whose idea is similar to the one illustrated in the previous section,
although the situation here is much more complicated since we need to extract
two independent perturbations with suitable properties. Their existence is
nontrivial since we are dealing with a general log-correlated field, and the
technical details are postponed to Section 4.3. The upshot of Section 4 is
Corollary 23 below, which yields a sufficient condition to the first part of
Theorem 2 in the form of a moment estimate of a weighted auxiliary positive
Gaussian multiplicative chaos measure.

###  4.1. Lemma on the density of the zeroes

The goal here is to provide a simple quantitative version of Lemma 12. In the
sequel, the symbol $\lesssim$ means smaller or equal to within a
multiplicative constant independent of $r=|z|$.

######  Lemma 16.

Let $z_{1},z_{2},\dots$ stand for the zeroes of a Blaschke product
$f:{\mathbb{D}}\to{\mathbb{D}}$ and let $0<\beta<1$. We have

$\sum_{k=1}^{\infty}(1-|z_{k}|)^{\beta}<\infty$

if and only if
$\displaystyle\int_{\mathbb{D}}(1-|z|^{2})^{\beta-2}\log(1/|\varphi(z)|)dA(z)<\infty$.

###### Proof.

Assume first that $\varphi$ is a finite Blaschke product. We set
$v(z)=(a-|z|^{2})^{\beta}$ with $0<\beta<1$ and first let the auxiliary
parameter $a$ satisfy $a>1$. Compute

$\Delta
v(z)=4\beta\big{(}\beta|z|^{2}-a\big{)}(a-|z|^{2})^{\beta-2}\asymp(a-|z|^{2})^{\beta-2}\quad\textrm{for}\;\,z\in{\mathbb{D}},$

where the implicit constants in $\asymp$ are independent of $a>1$. We note
that $v$ is smooth on $\overline{\mathbb{D}}$ up to the boundary, and so is
$z\mapsto\log(1/|\varphi(z)|)$ apart from its poles, and the latter function
vanishes on the boundary. Hence we may apply the Green’s formula and the fact
$\frac{1}{2\pi}\Delta\log|\varphi(z)|=\sum\limits_{z\in\mathbb{C};\;\varphi(z)=0}\delta_{z}$
on the unit disc to obtain

$\displaystyle\sum\limits_{k\geq 1}(a-|z_{k}|^{2})^{\beta}$
$\displaystyle\asymp$
$\displaystyle\frac{1}{2\pi}\int_{\mathbb{D}}(a-|z|^{2})^{\beta-2}\log(1/|\varphi(z)|)dA(z)+\frac{1}{2\pi}\int_{\partial
D}v(z)\nabla(\log(1/|\varphi(z)|))\cdot z,$

where $\cdot$ stands for the ${\mathbb{R}}^{2}$ inner product. By letting
$a\downarrow 1^{+}$ we obtain

$\sum\limits_{k\geq
1}(1-|z_{k}|^{2})^{\beta}\asymp\int_{\mathbb{D}}(1-|z|^{2})^{\beta-2}\log(1/|\varphi(z)|)dA(z).$

This same relation is then obtained for a general Blaschke product by applying
it first to a partial Blaschke product and using monotone convergence.
Finally, the lemma follows by noting that in our setup the series
$\sum\limits_{k\geq 1}(1-|z_{k}|^{2})^{\beta}$ and $\sum\limits_{k\geq
1}(1-|z_{k}|)^{\beta}$ converge simultaneously. ∎

######  Corollary 17.

The first part of Theorem 2 holds, if for all $\varepsilon>0$ and all $r$
close enough to $1$, we have for our random inner function $\varphi$

$\sup\limits_{|z|=r}\mathbb{E}\left[\log\left(\frac{1}{|\varphi(z)|}\right)\right]\lesssim(1-r)^{\frac{\gamma^{2}}{8}-\varepsilon}.$

To establish the first part of Theorem 2, we use a slightly weaker form of
Corollary 17. In order to state it, let us denote by $x(z)$ and $y(z)$
respectively the real and imaginary parts of $h(z)$, i.e.

$h(z)=\int_{{\mathbb{T}}}\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu(\theta)=x(z)+iy(z).$

We can write them explicitly in terms of the Poisson kernel and its conjugate:

$\begin{split}x(z)=\int_{{\mathbb{T}}}P_{z}(e^{i\theta})d\mu(\theta),\qquad&P_{z}(e^{i\theta})=\frac{1-|z|^{2}}{|z-e^{i\theta}|^{2}},\\\
y(z)=\int_{{\mathbb{T}}}Q_{z}(e^{i\theta})d\mu(\theta),\qquad&Q_{z}(e^{i\theta})=\frac{-2|z|\sin(\theta-\arg(z))}{|z-e^{i\theta}|^{2}}.\end{split}$

######  Corollary 18 (Upper bound for the density problem).

The first part of Theorem 2 holds if, for all $\varepsilon>0$,

$\mathbb{E}\left[\log\left(1+\frac{4x(z)}{(x(z)-1)^{2}}\right)\right]\lesssim(1-|z|)^{\frac{\gamma^{2}}{8}-\varepsilon}\qquad\textrm{\rm
for}\qquad z\in{\mathbb{D}}.$

This follows directly from Lemma 16 by writing

$\mathbb{E}\left[\log\left(\frac{1}{|\varphi(z)|}\right)\right]=\mathbb{E}\left[\log\left(\frac{|h(z)+1|}{|h(z)-1|}\right)\right]=\frac{1}{2}\mathbb{E}\left[\log\left(1+\frac{4x(z)}{(x(z)-1)^{2}+y(z)^{2}}\right)\right].$

###  4.2. Control of the log-singularity via perturbation

We first establish an auxiliary result in the spirit of Lemma 15, factorizing
out this time two independent Gaussian components. The idea is the same: by
establishing a lower bound in the derivatives of certain directions, we can
reduce the estimate with log-singularity to a relatively classical moment
estimate. Now that we parametrize the Gaussian perturbation in higher
dimensions, the estimation becomes considerably more complicated and we need
to study level sets of convex functions with suitable derivative bounds.

The proof of the following proposition is technical and will be postponed to
Section 4.3.

######  Proposition 19.

Consider a log-correlated Gaussian field $X$ on ${\mathbb{T}}$ with kernel

$K(\theta,\theta^{\prime})=-\log|\theta-\theta^{\prime}|+g(\theta,\theta^{\prime})$

with $g\in W^{s,2}({\mathbb{T}}^{2})$ for some $s>1$. There exist two
continuous and real functions $f_{1},f_{2}$ on ${\mathbb{T}}$ satisfying

$\inf_{t\in[0,2\pi)}\left(|f_{1}(\theta)|^{2}+|f_{2}(\theta)|^{2}\right)>0$

and a decomposition of $X$ into three independent Gaussian components,

$X(\theta)=V_{1}f_{1}(\theta)+V_{2}f_{2}(\theta)+\widetilde{X}(\theta)$

with $V_{1},V_{2}$ i.i.d. standard Gaussians and independent of the residual
field $\widetilde{X}$.

If we denote by $\mu$ and $\widetilde{\mu}$ the associated Gaussian
multiplicative chaos to $X$ and $\widetilde{X}$, then for $z\in{\mathbb{D}}$,

$x(z)=\int_{{\mathbb{T}}}\exp\left(\gamma V_{1}f_{1}(\theta)+\gamma
V_{2}f_{2}(\theta)-\frac{\gamma^{2}}{2}(f_{1}(\theta)^{2}+f_{2}(\theta)^{2})\right)P_{z}(\theta)d\widetilde{\mu}(\theta)$

where $P_{z}(\theta)$ is the Poisson kernel as before. Let $y_{1},y_{2}$
parametrize the Gaussians $V_{1},V_{2}$ and define the random function

(8) $u(y_{1},y_{2})=\int_{{\mathbb{T}}}\exp\left(\gamma
y_{1}f_{1}(\theta)+\gamma
y_{2}f_{2}(\theta)-\frac{\gamma^{2}}{2}(f_{1}(\theta)^{2}+f_{2}(\theta)^{2})\right)P_{z}(\theta)d\widetilde{\mu}(\theta).$

Especially, by the tower law we obtain

$\mathbb{E}\left[\log\left(1+\frac{4x(z)}{(x(z)-1)^{2}}\right)\right]\;=\;\frac{1}{2\pi}\mathbb{E}\left[\int_{{\mathbb{R}}^{2}}\log\left(1+\frac{4u(y_{1},y_{2})}{(u(y_{1},y_{2})-1)^{2}}\right)e^{-\gamma^{2}(y_{1}^{2}+y_{2}^{2})/2}dy_{1}dy_{2}\right].$

The key to the estimation of the above integral will be the following analytic
proposition.

######  Proposition 20.

Assume that $u:\mathbb{R}^{2}\to\mathbb{R}$ is positive, smooth, convex, and
satisfies $\Delta u\geq\kappa u$ for some constant $\kappa>0$ together with
$|Du|\leq Ku$ for some constant $K>0$.

  1. (i)

There is a constant $c_{0}<\infty$, independent of $u$, such that

$\int_{Q}\log\left(1+\frac{4u(y_{1},y_{2})}{(u(y_{1},y_{2})-1)^{2}}\right)dy_{1}dy_{2}\leq
c_{0}$

for every square $Q\subset\mathbb{R}^{2}$ of side length $1$.

  2. (ii)

The following integral is controlled by the value of $u(0,0)$:

$I\coloneqq\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\left[\log\left(1+\frac{4u(y_{1},y_{2})}{(u(y_{1},y_{2})-1)^{2}}\right)\right]e^{-\gamma^{2}(y_{1}^{2}+y_{2}^{2})/2}dy_{1}dy_{2}\leq
c_{1}(u(0,0)\wedge c_{2}).$

The constants $c_{1}$ and $c_{2}$ depend only on $\kappa,K$ and $\gamma$.

Before proving this result we first check that the function defined via (8)
satisfies the conditions of the above proposition.

######  Lemma 21.

The function $u$ defined in (8) almost surely satisfies the conditions of the
above proposition.

###### Proof.

Note first that $(y_{1},y_{2})\mapsto\exp(ay_{1}+by_{2})$ is convex, and sums
(or integrals of a parametrized family) of convex functions stays convex, so
$u$ is convex. Next, we may estimate

$\displaystyle\Delta u(y_{1},y_{2})$ $\displaystyle=$
$\displaystyle\gamma^{2}\int_{{\mathbb{T}}}\big{(}f_{1}(\theta)^{2}+f_{2}(\theta)^{2}\big{)}\exp\left(\gamma
y_{1}f_{1}(\theta)+\gamma
y_{2}f_{2}(\theta)-\frac{\gamma^{2}}{2}\big{(}f_{1}(\theta)^{2}+f_{2}(\theta)^{2}\big{)}\right)$
$\displaystyle\phantom{aaaaaaaaaaaaaaaaaa}\times
P_{z}(\theta)d\widetilde{\mu}(\theta)$ $\displaystyle\geq$ $\displaystyle
A\gamma^{2}u(y_{1},y_{2}),$

where
$A=\inf_{t\in[0,2\pi)}\big{(}|f_{1}(\theta)|^{2}+|f_{2}(\theta)|^{2}\big{)}>0$.
The upper bounds for the derivatives follow in a similar way, and finally the
positivity of $u$ is evident. ∎

We write hereafter $y=(y_{1},y_{2})\in\mathbb{R}^{2}$ for simplicity.

######  Lemma 22.

Assume that $u:\mathbb{R}^{2}\to\mathbb{R}$ is positive, of class $C^{2}$, and
satisfies $\Delta u\geq\kappa u$ with $\kappa>0$. For any ball
$B=\overline{B(y_{0},r)}\subset\mathbb{R}^{2}$, denote
$M_{u}(B)\coloneqq\max\limits_{y\in B}u(y)$ and
$m_{u}(B)\coloneqq\min\limits_{y\in B}u(y)$. Then

$M_{u}(B)\geq(1+\kappa r^{2}/4)m_{u}(B).$

###### Proof.

By translation we may assume that $y_{0}=0$. The assumption yields that
$\Delta(u(y)-A|y|^{2})\geq 0$ in $B$ for any $A\leq\kappa m_{u}/4$ . Thus the
subharmonic function $y\mapsto u(y)-A|y|^{2}$ achieves its maximum at the
boundary. In particular, for some $y_{1}\in\partial B$ we have
$u(y_{1})-Ar^{2}\geq u(0)$, which yields the claim. ∎

Intuitively, we are interested in the geometry around the level set $u(y)=1$,
where the singularity is present. All non-degenerate level sets are convex
Jordan curves by convexity of $u$, and the above lemma can be transformed into
an upper bound on the distance between levels sets of $1$ and $1\pm t$.

###### Proof of Proposition 20.

We first prove item (i). According to Lemma 22, our function $u$ is convex,
tends to infinity and is not constant in any non-empty open set. Denote
$m=\inf_{z\in{\mathbb{R}}^{2}}u(z)\geq 0$. The sublevel sets
$\Omega_{t}\coloneqq\\{y:u(y)<t\\}$ for $t>m$ are convex, non-empty and open
sets, increasing in $t$, and the level sets $S_{t}\coloneqq\\{y:u(y)=t\\}$
satisfy $\partial\Omega_{t}=S_{t}$. In turn, $S_{m}$ is either empty, a point,
or a closed line (segment).

Let $m<t<t_{2}$ and assume that $y_{0}\in\partial\Omega_{t}$. By convexity,
for given $r>0$, we may pick an open ball $B$ of radius $r/2$ such that
$B\subset\mathbb{C}\setminus\overline{\Omega}_{t}$ and $y_{0}\in\overline{B}$.
Lemma 22 implies that $\overline{B}\cap\partial\Omega_{t_{2}}\neq\emptyset$ as
soon as $t(1+\kappa r^{2}/16)\geq t_{2}$. Let us denote by
$A_{\varepsilon}\coloneqq\\{y\in\mathbb{C}:d(y,A)\leq\varepsilon\\}$ the
$\varepsilon$-fattening of a given set $A\subset\mathbb{C}$, we thus have
shown that $\partial\Omega_{t}$ is contained in
$(\partial\Omega_{t_{2}})_{\varepsilon}$ if $\varepsilon\geq
4\sqrt{\kappa^{-1}(t_{2}/t-1)}$. Especially, if $m\leq t_{1}<t_{2}$, as
$\Omega_{t_{2}}\setminus\Omega_{t_{1}}=\bigcup_{t_{1}\leq
t<t_{2}}\partial\Omega_{t}$, we deduce that

(9)
$\Omega_{t_{2}}\setminus\Omega_{t_{1}}\subset(\partial\Omega_{t_{2}})_{\varepsilon}\qquad\text{with}\qquad\varepsilon=4\sqrt{\kappa^{-1}(t_{2}/t_{1}-1)}.$

We now estimate the size of the intersection
$(\Omega_{t_{2}}\setminus\Omega_{t_{1}})\cap Q$. Assuming $\varepsilon<1/2$ we
obviously have

$(\partial\Omega_{t_{2}})_{\varepsilon}\cap
Q\subset\big{(}\partial(\Omega_{t_{2}}\cap 2Q)\big{)}_{\varepsilon},$

where $2Q$ is the twice dilated cube with the same center as $Q$. The domain
$\Omega_{t_{2}}\cap 2Q$ is convex, and its boundary has length at most $8$
(one can define a contraction by sending each point of $\partial(2Q)$ to its
nearest point in $\partial(\Omega_{t_{2}}\cap 2Q)$ and this shortens the
boundary length), and we deduce by (9) that

$|(\Omega_{t_{2}}\setminus\Omega_{t_{1}})\cap
Q|\;\leq\;4\Big{(}8\times\kappa^{-1/2}\sqrt{t_{2}/t_{1}-1}+\pi\Big{(}\kappa^{-1/2}\sqrt{t_{2}/t_{1}-1}\Big{)}^{2}\Big{)},$

which implies for $t\in(0,1/2)$ that

$|(\Omega_{1+t}\setminus\Omega_{1-t})\cap Q|\leq
80(\kappa^{-1/2}+\kappa^{-1})\sqrt{2t/(1-t)}\leq c_{1}\sqrt{t},$

where $c_{1}\coloneqq 160(\kappa^{-1/2}+\kappa^{-1})$. Previously we were
implicitly assuming that $m\leq 1/2$, but the analysis goes through with
obvious changes when $m\in(1/2,3/2)$, and the case $m\geq 3/2$ is not needed.

Finally, this enables us to estimate, separating the case $|u(y)-1|>1/2$ and
$|u(y)-1|\leq 1/2$,

$\begin{split}&\int_{Q}\log\left(1+\frac{4u(y)}{(u(y)-1)^{2}}\right)d^{2}y\\\
\leq{}&\log(1+6(1/2)^{-2})+\int_{(\Omega_{3/2}\setminus\Omega_{1/2})\cap
Q}\log\left(1+\frac{6}{(u(y)-1)^{2}}\right)d^{2}y\\\
\leq{}&\log(25)+\int_{0}^{1/2}\frac{c_{1}}{{2}}t^{-1/2}\log(1+6t^{-2})dt\eqqcolon
c_{0}.\end{split}$

For item (ii), let us first assume that $u(0)\eqqcolon a<c_{2}$, where
$c_{2}\leq e^{-4K}/2$ will be fixed later. Then we partition $\mathbb{R}^{2}$
into squares $Q_{m,n}\coloneqq(m,n)+[0,1)^{2}$, where
$(m,n)\in\mathbb{Z}^{2}$, and apply the estimate of item (i) on each of these
squares separately. We invoke the knowledge $|\nabla u|\leq Ku$, which implies
in the polar coordinates $|du/dr|\leq Ku$, and $u(y)\leq e^{K|y|}a$.
Especially $u(y)\leq 1/2$ inside the ball $B(0,\log(1/2a)/K)$.

We then consider separately the integrals over the domains $B(0,\log(1/2a)/K)$
and $\mathbb{C}\setminus B(0,\log(1/2a)/K)$. In the first domain,
$\log\big{(}1+\frac{4u(y)}{(u(y)-1)^{2}}\big{)}\lesssim u(y)$ and we bound the
integral by

$\displaystyle
C\frac{1}{2\pi}\int_{B(0,\log(1/2a)/K)}u(y)e^{-\gamma^{2}|y|^{2}/2}d^{2}y\;\lesssim\;a\int_{0}^{\infty}re^{Kr}e^{-\gamma^{2}r^{2}/2}dr$
$\displaystyle\lesssim$ $\displaystyle
a\int_{0}^{\infty}re^{(1+K/\gamma)r}e^{-r^{2}/2}dr\;\lesssim\;e^{\frac{(1+K/\gamma)^{2}}{2}}a.$

When dealing with the second domain, let $l\geq\log(1/2a)/K\geq 4$ be an
integer. We apply part (i) on the square $Q_{m,n}$ with $Q_{m,n}\cap\partial
B(0,l)\neq\emptyset$, and see that the integral over $Q_{m,n}$ is at most

$c_{0}e^{-\gamma^{2}(l-\sqrt{2})^{2}/2}.$

Every square $Q_{m,n}$ intersects some $\partial B(0,l)$ with an integer $l$,
and for each fixed $l\neq 0$, the number of such squares is at most $250861l$.
Summing up we obtain the following upper bound in the second case:

$\begin{split}&c_{0}\sum\limits_{l\geq\log(1/2a)/K,\leavevmode\nobreak\
l\in\mathbb{N}}250861le^{-\gamma^{2}(l-\sqrt{2})^{2}/2}\\\
\leq{}&250861c_{0}\int_{\log(1/2a)/K-1-\sqrt{2}}^{\infty}(x+\sqrt{2})e^{-\gamma^{2}x^{2}/2}dx\\\
\leq{}&250861c_{0}\int_{\log(1/2a)/K-1-\sqrt{2}}^{\infty}2xe^{-\gamma^{2}x^{2}/2}dx\\\
\lesssim{}&2\times
250861c_{0}e^{-\gamma^{2}(\log(1/2a)/K-1-\sqrt{2})^{2}/2}\\\
\lesssim{}&c_{0}a\end{split}$

where the first inequality holds as
$x\mapsto(x+\sqrt{2})e^{-\gamma^{2}x^{2}/2}$ is decreasing on the range of
integration, and the last one e.g. if $-\log(2a)\geq 6+2K^{2}/\gamma^{2}$. All
the needed constaints are clearly satisfied for $a<c_{2}<1$ by choosing
$c_{2}$ small enough depending on $K$ and $\gamma.$

In the case $a\geq c_{2}$, it suffices to show that $I$ is bounded by a
constant independent of $a$. This easily follows from item (i) by again
summing over all $Q_{m,n}$ with $(m,n)\in\mathbb{Z}^{2}$. ∎

The upshot of this section is:

######  Corollary 23.

The first part of Theorem 2 holds if, for all $z\in\mathbb{D}$ and
$\varepsilon>0$,

$\mathbb{E}\left[\widetilde{x}(z)\wedge
1\right]\lesssim(1-|z|)^{\frac{\gamma^{2}}{8}-\varepsilon}$

where with the notations in the beginning of this section,

$\widetilde{x}(z)=\int_{{\mathbb{T}}}P_{z}(\theta)d\widetilde{\mu}(\theta).$

###### Proof.

Combine Corollary 18, Equation (8) and item (ii) of Proposition 20. ∎

###  4.3. Existence of a rank two independent Gaussian summand

We now prove Proposition 19 by showing that one may actually choose $f_{1}$
and $f_{2}$ to be two trigonometric polynomials. We start first by considering
suitable compact perturbations of the identity operator.

######  Lemma 24.

Assume that $T$ is compact and self-adjoint on a separable Hilbert space $H$
and $\mathrm{Id}+T\geq 0$. Then there are vectors
$\varphi_{1},\dots,\varphi_{l_{0}}$ with the following property: for a given
vector $u\in H$ there exists $\varepsilon>0$ such that

$T-\varepsilon u\otimes u\geq 0.$

if and only if

$\langle u,\varphi_{j}\rangle=0\qquad\textrm{for}\qquad j=1,\dots,l_{0}.$

###### Proof.

Let $\\{\varphi_{n}\\}_{n=1}^{\infty}$ be an orthonormal basis of $H$
consisting of eiegnvectors of $T$, ordered so that the eigenvectors that
correspond to eigenvalue $-1$ are listed first, and write the spectral
decomposition

$T=-\sum_{j=1}^{l_{0}}\varphi_{j}\otimes\varphi_{j}+\sum_{j>l_{0}}\lambda_{j}\varphi_{j}\otimes\varphi_{j}.$

Here we used the compactness of the operator $T$ to deduce that $l_{0}$ may be
taken finite, and we also see that there is $\varepsilon>0$ such that
$\lambda_{j}>\varepsilon-1$ for all $j>l_{0}.$ Assume then that $u\in H$ is
normalized and satisfies the stated condition so that
$u=\sum_{j>l_{0}}u_{j}\varphi_{j}$ with $\sum_{j>l_{0}}u_{j}^{2}=1$. If $x\in
H$ is arbitrary, $x=\sum_{j=1}^{\infty}x_{j}\varphi_{j}$, we may compute

$\displaystyle\langle x,(\mathrm{Id}+T-\varepsilon u\otimes u)x\rangle$
$\displaystyle=$
$\displaystyle\sum_{j>l_{0}}(1+\lambda_{j})x_{j}^{2}-\varepsilon(\sum_{j>l_{0}}u_{j}x_{j})^{2}$
$\displaystyle\geq$
$\displaystyle\varepsilon\sum_{j>l_{0}}x_{j}^{2}-\varepsilon(\sum_{j>l_{0}}x_{j}^{2})(\sum_{j>l_{0}}u_{j}^{2})=0.$

In turn, the necessity of the condition is seen by observing that if
$b=\sum_{j=1}^{\infty}b_{j}\varphi_{j}$, with $b_{j_{0}}\neq 0$ for some
$1\leq j_{0}\leq l_{0}$, we obtain
$\langle\varphi_{j_{0}},(\mathrm{Id}+T-\varepsilon b\otimes
b)\varphi_{j_{0}}\rangle=-b_{j_{0}}^{2}<0$. ∎

The following example shows that it is not always enough to consider
perturbations with one base element. In the context of Proposition 19 this
means that choosing $f_{1}$ to be simply a trigonometric monomial will not
always suffice.

######  Example 25.

Let $(e_{j})_{j\geq 1}$ be an orthonormal basis of $H$, and set
$a\coloneqq\sum_{j=1}^{\infty}2^{-j/2}e_{j}$. Then $\mathrm{Id}-a\otimes a\geq
0$. However, for all $n\geq 1$ and $\varepsilon>0$ we have
$\mathrm{Id}-a\otimes a-\varepsilon e_{n}\otimes e_{n}\not\geq 0$.

In our next auxiliary result, we generalize Lemma 24 to cover operators of
form $C+A$, where the compact operator $C$ is assumed to dominate $A$ in a
suitable sense (i.e. $A$ is compact with respect to $C$, see condition (10)
below).

######  Lemma 26.

Assume that $C,A$ are compact (symmetric) and self-adjoint operators so that
both $C\geq 0$ and $C+A\geq 0$. Assume also that the kernel
$M\coloneqq\mathrm{Ker}(C)$ is finite-dimensional, and denote by $P$ the
orthogonal projection onto $M$. We write $C_{1}\coloneqq C+P$ and assume that

(10) $C_{1}^{-1/2}AC_{1}^{-1/2}\qquad\textrm{extends to a compact operator
on}\leavevmode\nobreak\ H.$

Then, there are elements $v_{1},\dots,v_{l_{0}}\in H$ such that if $u\in H$
satisfies $\langle u,v_{j}\rangle=0$ for $j=1,\dots,l_{0},$ then

$C+A-\varepsilon C_{1}^{1/2}u\otimes C_{1}^{1/2}u\geq 0$

for small enough $\varepsilon>0.$

###### Proof.

We may actually assume that $M=\\{0\\}$ by replacing $C$ by $C_{1}$ and $A$ by
$A-P$. The claim then follows easily by applying the previous lemma with
$T=C_{1}^{-1/2}AC_{1}^{-1/2}$. ∎

######  Corollary 27.

In the situation of Lemma 26, let $\\{e_{j}\\}$ denote an orthonormal basis of
$H$ consisting of eigenvectors of $C_{1}$. Then for any $E\subset{\mathbb{N}}$
with $|E|\geq l_{0}+1$ there is a nontrivial linear combination $b=\sum_{j\in
E}a_{j}e_{j}$ and $\varepsilon>0$ so that

$C+A-\varepsilon b\otimes b\geq 0.$

###### Proof.

Simply note that if $u$ is a linear combination of eigenvectors of $C_{1}$,
then $b=C_{1}^{1/2}u$ is also. ∎

We then apply the previous observations to our situation.

######  Lemma 28.

Assume that $C$ is the covariance of the canonical field on ${\mathbb{T}}$ and
let $A\in W^{2,2}({\mathbb{T}}^{2})$ be a symmetric (real-valued) function
such that $C+A\geq 0$. We shall also denote by $C,A$ the corresponding
operators on $L^{2}({\mathbb{T}})$. Then there are nontrivial finite
trigonometric polynomials $f_{1},f_{2}$ such that

$C+A-f_{1}\otimes f_{1}-f_{2}\otimes f_{2}\geq 0$

and

$\inf_{\theta\in[0,2\pi)}\left(|f_{1}(\theta)|^{2}+|f_{2}(\theta)|^{2}\right)>0.$

###### Proof.

Let us denote by $(e_{j})_{j\geq 0}$ the standard trigonometric basis of
$L^{2}$, where $e_{0}$ is constant, and $e_{2j-1},e_{2j}$ are the sine and
cosine functions with frequency $j$ for $j\geq 1$. In this basis $C,C_{1}$ are
the diagonal operators

$\displaystyle C$ $\displaystyle=$ $\displaystyle{\rm
diag\,}(0,1,1,1/2,1/2,1/3,1/3,\dots),$ $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle{\rm diag\,}(1,1,1,1/2,1/2,1/3,1/3,\dots)\;\eqqcolon\;{\rm
diag\,}(m),$

and $P=e_{0}\otimes e_{0}$ is one-dimensional. Especially, $C_{1}$ is the
Fourier-multiplier $T_{m}$, where $m$ is the above defined vector, and we see
immediately that $C:W^{s,2}(T)\to W^{s+1,2}(T)$ is an isomorphism. The powers
$C^{\alpha}$ are naturally defined. Lemma 26 applies as soon as we verify that
(10) is satisfied. This follows e.g. by checking that
$C_{1}^{-1/2}AC_{1}^{-1/2}$ is Hilbert-Schmidt, which is clearly equivalent to
$(1-\Delta_{x})^{1/4}(1-\Delta_{y})^{1/4}A\in L^{2}({\mathbb{T}}^{2})$.
Writing this in terms of the Fourier coefficients and using Cauchy-Schwarz, we
see that this is implied by the fact that $A\in W^{1,2}({\mathbb{T}}^{2}).$

Let us denote by $\varphi_{1},\dots,\varphi_{l}$ an orthonormal basis of the
eigenspace correponding to eigenvalue $-1$ of the operator
$C_{1}^{-1/2}AC_{1}^{-1/2}$. Then the condition for a trigonometric polynomial
$f$ ensuring that $C+A-\varepsilon f\otimes f\geq 0$ for some $\varepsilon>0$
can be written in the form

$\langle f,C_{1}^{-1/2}\varphi_{j}\rangle=0,\qquad j=1,\dots l,$

where the computation is a priori formal. However, the element
$C_{1}^{-1/2}\varphi_{j}$ belongs to $L^{2}({\mathbb{T}})$ since we check as
above that $C_{1}^{-1}AC_{1}^{-1}$ extends to a bounded operator on $L^{2}$
(since $A\in W^{2,2}({\mathbb{T}}^{2})$), and hence we obtain

$C_{1}^{-1/2}\varphi_{j}=C_{1}^{-1/2}\big{(}-C_{1}^{-1/2}AC_{1}^{-1/2}\varphi_{j}\big{)}=-C_{1}^{-1}AC_{1}^{-1}(C_{1}^{1/2}\varphi_{j})\in
L^{2}({\mathbb{T}}).$

Let us denote $C_{1}^{-1/2}\varphi_{j}\eqqcolon\psi_{j}$ for $j=1,\dots,l$.
This means that $C+A-\varepsilon f\otimes f\geq 0$ for some $\varepsilon>0$ as
soon as the trigonometric polynomial $f$ satisfies

(11) $\langle f,\psi_{j}\rangle=0,\qquad j=1,\dots l.$

We still need to find two trigonometric polynomials $f_{1}$ and $f_{2}$, such
that they both satisfy (11) and without common zeroes. For this purpose, we
first fix $f_{1}$ to be any non-trivial trigonometric polynomial satisfying
(11). This can obviously be done by choosing the degree of $f_{1}$ to be at
least $l+1$. The polynomial $f_{1}$ has only finitely many zeroes. Using a
perturbative argument and induction on the number of zeroes, it clearly
suffices to find $f_{2}$ such that it does vanish in a given point
$\theta_{0}$ and it satisfies (11). For any sequence $y=(y_{n})_{n\geq 0}$
denote by $P_{N}y$ the finite sequence $P_{N}y\coloneqq(y_{n})_{0\leq n\leq
N}$. A trigonometric polynomial $f_{2}=\sum_{j=0}^{N}c_{j}e_{j}$ vanishes at
$\theta_{0}$ exactly when $\langle f_{2},P_{N}g_{0}\rangle=0$, where

$g_{0}\coloneqq(1,\sin(\theta_{0}),\cos(\theta_{0}),\sin(2\theta_{0}),\dots).$

Thus the choice of $f_{2}$ as a trigonometric polynomial is not possible if
and only if

$P_{N}g_{0}\in{\rm
span\,}\\{P_{N}\psi_{1},\dots,P_{N}\psi_{l}\\}\qquad\textrm{ for all}\;\;N.$

However, then an elementary argument implies that $g_{0}\in{\rm
span\,}\\{\psi_{1},\dots,\psi_{l}\\}$, which is impossible since
$g_{0}\notin\ell^{2}.$ Now the positivity of both
$C+A-\varepsilon_{1}f_{1}\otimes f_{1}$ and $C+A-\varepsilon_{2}f_{2}\otimes
f_{2}$ implies

$C+A-\frac{1}{2}\varepsilon_{1}f_{1}\otimes
f_{1}-\frac{1}{2}\varepsilon_{2}f_{2}\otimes f_{2}\geq 0,$

and this completes the proof. ∎

##  5\. Proof of the upper bound on the density of zeroes

In this section we verify the sufficient condition in Corollary 23, which
implies the first part of Theorem 2, and it also automatically implies the
general form of Theorem 1. Since

(12) $\mathbb{E}\left[\widetilde{x}(z)\wedge
1\right]\lesssim\mathbb{E}\left[\widetilde{x}(z)^{p}\right],$

for all $p\in(0,1)$, it suffices to establish a $p$-moment bound on
$\widetilde{x}(z)$ with $p<1$. We only need the fact that $\widetilde{x}(z)$
is generated by a general log-correlated Gaussian field in the rest of the
proof, so some notations are deliberately not strict (the proofs work for any
Gaussian multiplicative chaos measures).

######  Lemma 29.

With the assumptions of Theorem 1 we have for any $p\in(0,1)$ the estimate

$\mathbb{E}\left[\widetilde{x}(z)^{p}\right]\lesssim(1-|z|)^{p(1-p)\gamma^{2}/2}.$

###### Proof.

We denote by $A_{\widetilde{\mu},\varepsilon}(\theta)$ stands for the average
of the Gaussian multiplicative chaos measure $d\widetilde{\mu}$ over the
interval $[\theta-\varepsilon,\theta+\varepsilon]$. Let us consider our
Gaussian field $\widetilde{X}$ on the interval $\theta\in(-1/2,1/2)$. Let
$X_{1}$ be an exactly scaling field on $(-1/2,1/2)$. At this point we refer to
Fact 6 in Section 2 for the properties of the exactly scaling field. For the
chaos $\mu_{1}$ constructed from the exactly scaling field $X_{1}$ it holds
that

$\mu_{1}(-\varepsilon,\varepsilon)\sim
2\varepsilon^{1+\gamma^{2}/2}\exp\big{(}\gamma
N(0,\log{1/\varepsilon})\big{)}\mu^{\prime}_{1}(-1/2,1/2),$

where $\mu^{\prime}_{1}$ is a copy of $\mu_{1}$ and the normal variable
$N(0,\log{1/\varepsilon})$ is independent of $\mu^{\prime}_{1}$. This
distributional equality yields immediately that
$\mathbb{E}\left[(\mu_{1}(-\varepsilon,\varepsilon))^{p}\right]=c_{p}\varepsilon^{p(1+\gamma^{2}(1-p)/2)}$
(use Fact 5 in Section 2). By the reasoning in Remark 8 we deduce the
corresponding upper bound for $\widetilde{\mu}$, i.e.

(13)
$\mathbb{E}\left[(\widetilde{\mu}(-\varepsilon,\varepsilon))^{p}\right]\lesssim\varepsilon^{p(1+\gamma^{2}(1-p)/2)}.$

To obtain the statement of the lemma we may assume (e.g. by Harnack) that
$z=r\in(0,1)$, where $1-r=2^{-k}$ for an integer $k\geq 2$. Let us denote
$q:=p(1+\gamma^{2}(1-p)/2).$ By a standard property of the Poisson extension
(that can be read directly from the kernel) we may upper bound
$\widetilde{x}(r)$ by the convex combination

(14) $\widetilde{x}(r)\lesssim
2^{-k}\widetilde{\mu}({\mathbb{T}})+\sum_{j=0}^{k}2^{-j}A_{\widetilde{\mu},2^{j-k}}(\theta).$

In view of this and (13) we may estimate (use $q<2p$ for $p\in(0,1)$)

$\mathbb{E}\left[\widetilde{x}(r)^{p}\right]\lesssim
2^{-kp}\mathbb{E}\left[\widetilde{\mu}({\mathbb{T}})^{p}\right]+\sum_{j=0}^{k}2^{-jp}2^{(j-k)q}2^{(k-j)p}\lesssim
2^{k(p-q)}\lesssim(1-r)^{q-p},$

which yields the claim by the definition of $q$. ∎

######  Corollary 30.

For all $\varepsilon>0$ and $z\in\mathbb{D}$,

$\mathbb{E}\left[\widetilde{x}(z)\wedge
1\right]\lesssim(1-|z|)^{\frac{\gamma^{2}}{8}-\delta},$

where $\delta>0$ is arbitrary.

###### Proof.

This follows simply by choosing $p=1/2$ in Lemma 29. ∎

This completes the proof for the first part of Theorem 2 in view of (12).

We end this section by presenting an alternative to Corollary 30 that is based
on the following known result:

######  Lemma 31.

Let $Y$ be a log-correlated Gaussian field on the unit circle ${\mathbb{T}}$
and $\mu_{Y}$ the associated Gaussian multiplicative chaos measure with
parameter $\gamma\in(0,\sqrt{2}]$. Then for all $0<p<1$ the quantity

$\sup_{z\in{\mathbb{T}}}\mathbb{E}\left[\left(\int_{{\mathbb{T}}}\frac{1}{|z-e^{i\theta}|^{s}}d\mu_{Y}(\theta)\right)^{p}\right]$

is finite if and only if $s<s(p)=1+\frac{\gamma^{2}}{2}(1-p)$.

This result is known in the physics literature on $2d$-Conformal Field Theory
as the extended Seiberg bound, see [DKRV16, Lemma 3.10] for the case on the
Riemann sphere. For the reader’s convenience we include below in Appendix A a
self-contained proof of this fact.

###### Second proof of Corollary 30.

The argument below does not depend on the log-correlated Gaussian field so let
$\mu$ be a general Gaussian multiplicative chaos measure on ${\mathbb{T}}$.
Since $|z-e^{i\theta}|^{2}\geq(1-|z|)^{2-s}|z-e^{i\theta}|^{s}$ we obtain

$\begin{split}\mathbb{E}[x(z)^{p}]=\mathbb{E}\left[\left(\int_{{\mathbb{T}}}\frac{1-|z|^{2}}{|z-e^{i\theta}|^{2}}d\mu(\theta)\right)^{p}\right]\lesssim(1-|z|^{2})^{(s-1)p}\mathbb{E}\left[\left(\int_{{\mathbb{T}}}\frac{1}{|z-e^{i\theta}|^{s}}d\mu(\theta)\right)^{p}\right].\end{split}$

By Lemma 31, the last expectation has a bound independent of $z$ if
$s<1+\frac{\gamma^{2}}{2}(1-p)$. By choosing $p=\frac{1}{2}$ and letting $s$
approach $1+\frac{\gamma^{2}}{2}(1-p)$ from below we obtain, for all
$\varepsilon>0$,

$(1-|z|^{2})^{(s-1)p}\leq(1-|z|^{2})^{\frac{\gamma^{2}}{8}-\varepsilon}.$

∎

##  6\. Proof of the lower bound on the density of zeroes

We now prove the second part of Theorem 2. Recall that
$A_{\mu,\varepsilon}(\theta)$ stands for the average of the Gaussian
multiplicative chaos measure $d\mu$ over the interval
$[\theta-\varepsilon,\theta+\varepsilon]$. We start by recalling a useful
result on the multifractal spectrum of chaos measure estimating the size of
the set of points $e^{i\theta}\in{\mathbb{T}}$ such that $A_{\mu,1-r}(\theta)$
is (roughly speaking) of order $(1-r)^{o(1)}$. According to [Ber23, Theorem
3.1] (see also an earlier discussion of multifractal spectra in [RV14, Section
4, especially equation (4.6)]) with $d=\alpha=1$ it holds that

${\rm dim}_{\mathcal{H}}(G_{0})=1-\frac{\gamma^{2}}{8},\quad\text{where}\quad
G_{0}\coloneqq\left\\{\theta\in[0,2\pi)\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\ \lim_{r\to 1^{-}}\frac{\log
A_{\mu,1-r}(\theta)}{\log(1-r)}=0\right\\}.$

Fix $\delta>0$. Since $G_{0}\subset\bigcup_{k\geq 1}G_{k}$ with

$G_{k}=\left\\{\theta\in[0,2\pi)\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\ \frac{\log
A_{\mu,1-r}(\theta)}{\log(1-r)}\in(-\delta,\delta)\quad\text{for all}\quad
r\in(1-1/k,1)\right\\},$

we may pick $r_{0}\in(1/2,1)$ so that

$\text{dim}_{\mathcal{H}}(G)\geq 1-\delta/2-\frac{\gamma^{2}}{8}$

with

$\quad G\coloneqq\left\\{\theta\in[0,2\pi)\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\ \frac{\log
A_{\mu,1-r}(\theta)}{\log(1-r)}\in(-\delta,\delta)\quad\text{for all}\quad
r\in(r_{0},1)\right\\}.$

We denote for given $\varepsilon>0$ the $\varepsilon$-thickening of the set
$G$ by $G_{\varepsilon}$, i.e.
$G_{\varepsilon}\coloneqq\\{\theta\leavevmode\nobreak\ |\leavevmode\nobreak\
{\rm d\,}(\theta,G)\leq\varepsilon\\}$. By standard estimates for the
Hausdorff content (see [Mat95, Chapter 5.5]), we then have the following lower
bound for the Lebesgue measure of the $\varepsilon$-thickening
$G_{\varepsilon}$, that

(15) $|G_{\varepsilon}|\gtrsim\varepsilon^{\gamma^{2}/8+\delta}.$

We then define the random subset $H\subset{\mathbb{D}}$ by setting

$H\coloneqq\\{re^{i\theta}\leavevmode\nobreak\ |\leavevmode\nobreak\
r\in(r_{0},1),\quad\theta\in G_{1-r}\\}$

and claim that

(16) $c_{0}(1-|z|)^{\delta}\leq x(z)\leq
c_{1}(1-|z|)^{-\delta}\quad\text{for}\quad z\in H,$

where the constants $c_{0},c_{1}$ are independent of $z$ but possibly random.
The lower bound is immediate, since if $z=re^{i\theta}\in H$, then there is
$\theta^{\prime}\in G$ with $|\theta^{\prime}-\theta|\leq 1-r$ and hence

$x(z)\geq
cA_{\mu,2(1-r)}(\theta)\geq(c/2)A_{\mu,1-r}(\theta^{\prime})\geq\frac{c}{2}(1-r)^{\delta}=\frac{c}{2}(1-|z|)^{\delta}.$

For the upper bound we proceed as in (14) and estimate the action of Poisson
kernel at $z^{\prime}\coloneqq re^{i\theta^{\prime}}$ by convex combinations
of the averages $A_{\mu,2^{k}(1-r)}(\theta^{\prime})$, where $k=0,1,\dots$
satisfies $2^{k}(1-r)\leq 1$. Using additonally the fact that $x$ is positive
harmonic function, Harnack’s inequality allows us to estimate

$\begin{split}x(z)&\;\;\leq\;\;c^{\prime}x(z^{\prime})\;\;\lesssim\;\;c^{\prime}\sum_{2^{k}(1-r)\;\leq
1}2^{-2k}A_{\mu,2^{k}(1-r)}(\theta^{\prime})\\\
&\;\;\leq\;\;c^{\prime}\sum_{2^{k}(1-r)\leq
1-r_{0}}2^{-2k}A_{\mu,2^{k}(1-r)}(\theta^{\prime})+c^{\prime}(1-r_{0})^{-1}\mu({\mathbb{T}})\\\
&\;\;\lesssim\;\;c^{\prime}(1-|z|)^{-\delta}+c^{\prime\prime},\end{split}$

where $c^{\prime\prime}$ is a random constant. This yields (16).

In order to control the contribution of the imaginary part $y$ we will use a
simple deterministic argument that relies on the fact that when the averages
of the measure are in control, then the imaginary part cannot be too large
either. Assume that $z=re^{i\theta}\in H$. Since the conjugate kernel has the
upper bound $\lesssim\max(|e^{i\theta}-z|^{-1},(1-r)^{-1})$ and we have

$\mu(\\{\theta^{\prime}\,\big{|}\,|\theta^{\prime}-\theta|\leq(1-r)2^{k}\\})=(1-r)2^{k}A_{\mu,2^{k}(1-r)}(\theta),$

it follows by considering dyadic annuli around $\theta$ such that the inner
radii are $(1-r)2^{k}$, with $k=0\ldots$, and the interval
$(\theta-(1-r),\theta+(1-r))$ separately, that

$\displaystyle|y(z)|$ $\displaystyle\lesssim$ $\displaystyle
c_{r_{0}}\mu({\mathbb{T}})+\sum_{2^{k}(1-r)\leq
1-r_{0}}A_{\mu,2^{k}(1-r)}(\theta)\;\leq\;c+c^{\prime}\log\left(\frac{1-r_{0}}{1-r}\right)(1-|z|)^{-\delta}$
$\displaystyle\lesssim$ $\displaystyle c_{2}(1-|z|)^{-2\delta},$

where $c_{2}$ is a random constant.

We are now ready to conclude the desired lower bound. For $z\in H$ it holds
that $\log|\varphi(z)|\gtrsim(1-|z|)^{-(\delta+4\delta)}$. By (15) we obtain
that

$\int_{|z|=r}-\log|\varphi(z)|d|z|\;\gtrsim\;(1-r)^{(\frac{\gamma^{2}}{8}+\delta)}(1-r)^{(5\delta)}.$

Since $6\delta$ can be chosen arbitrary small this implies the claim by Lemma
16.

##  Appendix A Proof of Lemma 31

It is clearly enough to estimate the moment for $z=1$. Consider the arc
$I=[e^{-i\pi/8},e^{i\pi/8}]\subset{\mathbb{T}}$ and $0<p<1$. By subadditivity,

$\displaystyle\mathbb{E}\left(\int_{{\mathbb{T}}}\frac{1}{|1-e^{i\theta}|^{s}}d\mu_{Y}(\theta)\right)^{p}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{I}\frac{1}{|1-e^{i\theta}|^{s}}d\mu_{Y}(\theta)\right)^{p}+\mathbb{E}\left(\int_{{\mathbb{T}}\setminus
I}\frac{1}{|1-e^{i\theta}|^{s}}d\mu_{Y}(\theta)\right)^{p}$
$\displaystyle\lesssim$
$\displaystyle\mathbb{E}\left(\int_{I}\frac{1}{|\theta|^{s}}d\mu_{Y}(\theta)\right)^{p}\;+\;C(p,\gamma),$

where $C(p,\gamma)<\infty$ by Fact 5.

By Kahane’s convexity inequality (see Fact 7 and the discussions thereafter),
we can trade $Y$, restricted to $I$, for the exact scaling log-correlated
field $\tilde{Y}$ with covariance kernel
$K(\theta,\theta^{\prime})=\log\frac{1}{|\theta-\theta^{\prime}|}$,
$\theta,\theta^{\prime}\in[-\frac{\pi}{8},\frac{\pi}{8}].$ In order to use the
good scaling properties of the field $\tilde{Y}$, consider a dyadic tiling of
the interval $[-\pi/8,\pi/8]$ by denoting
$Q_{k}\coloneqq[-2^{-k}\pi/8,2^{-k}\pi/8]$, and $R_{k}\coloneqq Q_{k}\setminus
Q_{k+1}$ for $k\geq 0$. We recall that the exact scaling property states that
for all $0<r<1$, in law,

$(\tilde{Y}(x+ry))_{y\in J}=(\tilde{Y}(x+y))_{y\in J}+\sqrt{-\log r}N$

for any interval $J\subset{\mathbb{R}}$ with $|J|<1$, where $N$ is a standard
Gaussian independent of everything else, see Fact 6. Using the relation above
with $x=0$, $J=Q_{k}$ and $r=\frac{1}{2}$, with a $\varepsilon$-regularization
by any scale-invariant mollifier, we get that for all $k\geq 0$,

(17)
$\begin{split}&\mathbb{E}\left[\left(\int_{R_{k+1}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]\\\
={}&\lim\limits_{\varepsilon\to
0}\mathbb{E}\left[\left(\int_{R_{k+1}}\frac{1}{|\theta|^{s}}e^{\gamma\tilde{Y}_{\varepsilon}(\theta)-\frac{\gamma^{2}}{2}\mathbb{E}\left[\tilde{Y}_{\varepsilon}(\theta)^{2}\right]}d\theta\right)^{p}\right]\\\
={}&\lim\limits_{\varepsilon\to
0}\mathbb{E}\left[\left(\int_{R_{k}}\frac{1}{|\theta/2|^{s}}e^{\gamma\tilde{Y}_{\varepsilon/2}(\theta/2)-\frac{\gamma^{2}}{2}\mathbb{E}\left[\tilde{Y}_{\varepsilon/2}(\theta/2)^{2}\right]}d\left(\frac{\theta}{2}\right)\right)^{p}\right]\\\
\stackrel{{\scriptstyle\ast}}{{=}}{}&2^{sp-p}\lim\limits_{\varepsilon\to
0}\mathbb{E}\left[\left(\int_{R_{k}}\frac{1}{|\theta|^{s}}e^{\gamma\tilde{Y}_{\varepsilon}(\theta)+\gamma\sqrt{\log
2}N-\frac{\gamma^{2}}{2}\mathbb{E}\left[\tilde{Y}_{\varepsilon}(\theta)^{2}\right]-\frac{\gamma^{2}}{2}\log
2}d\theta\right)^{p}\right]\\\
={}&2^{sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2}\lim\limits_{\varepsilon\to
0}\mathbb{E}\left[\left(\int_{R_{k}}\frac{1}{|\theta|^{s}}e^{\gamma\tilde{Y}_{\varepsilon}(\theta)-\frac{\gamma^{2}}{2}\mathbb{E}\left[\tilde{Y}_{\varepsilon}(\theta)^{2}\right]}d\theta\right)^{p}\right]\\\
={}&2^{sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2}\mathbb{E}\left[\left(\int_{R_{k}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right],\end{split}$

where in step $(\ast)$ we used the scaling relation.

The careful reader might has perhaps observed, that we have above actually
treated only the subcritical chaos, i.e. $\gamma<\sqrt{2}$. In order to extend
to the case of critical chaos, we simply use the definition via the so-called
Seneta-Heyde normalization, where one defines the critical chaos
$\mu^{\gamma_{c}}_{Y}$ as the weak-$\star$ limit in probability

$\mu^{\gamma_{c}}_{Y}=\lim\limits_{\varepsilon\to
0}\sqrt{-\log\varepsilon}\mu_{\varepsilon}^{\gamma_{c}},$

where $\mu_{\varepsilon}^{\gamma_{c}}$ is the $\varepsilon$-regularized
measure with parameter $\gamma_{c}=\sqrt{2}$,

$d\mu^{\gamma_{c}}_{Y}(\theta)=\lim\limits_{\varepsilon\to
0}e^{\gamma_{c}Y_{\varepsilon}(\theta)-\frac{\gamma_{c}^{2}}{2}\mathbb{E}\left[Y_{\varepsilon}(\theta)^{2}\right]}d\theta.$

Via this definition one immediately checks that (17) remains valid also for
$\gamma=\sqrt{2}$. For other equivalent definitions and a review on the
critical Gaussian chaos measures, see [Pow20].

Notice that the exponent $sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2$ is negative
if and only if $s<s(p)=1+\frac{\gamma^{2}}{2}(1-p)$. Similar scaling relation
holds between $Q_{k+1}$ and $Q_{k}$. When

$\mathbb{E}\left[\left(\int_{[-\pi/8,\pi/8]}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]<\infty,$

we have, since $Q_{k+1}\subset Q_{k}$,

$\begin{split}\mathbb{E}\left[\left(\int_{Q_{1}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]&=2^{sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2}\mathbb{E}\left[\left(\int_{Q_{0}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]\\\
&>2^{sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2}\mathbb{E}\left[\left(\int_{Q_{1}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right],\end{split}$

in which case it is necessary to have $s<s(p)=1+\frac{\gamma^{2}}{2}(1-p)$. On
the other hand, if this is satisfied, then subadditivity yields for any
$0<p<1$,

$\mathbb{E}\left[\left(\int_{[-\pi/8,\pi/8]}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]\leq\sum\limits_{k=0}^{\infty}\mathbb{E}\left[\left(\int_{R_{k}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]=C_{0}\sum\limits_{k=0}^{\infty}2^{(sp-p(1+\gamma^{2}/2)+p^{2}\gamma^{2}/2)k}$

where
$C_{0}=\mathbb{E}\left[\left(\int_{R_{0}}\frac{1}{|\theta|^{s}}d\mu_{\tilde{Y}}(\theta)\right)^{p}\right]<\infty$
since the Gaussian chaos measures in $1d$ (even in the critical case) have
finite $p$-moments for $p<1$. ∎

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