# Directed harmonic currents near
non-hyperbolic linearized singularities

Zhangchi Chen

###### Abstract

Let $(\mathbb{D}^{2},\mathscr{F},\\{0\\})$ be a singular holomorphic foliation
on the unit bidisc $\mathbb{D}^{2}$ defined by the linear vector field

$z\,\frac{\partial}{\partial z}+\lambda\,w\,\frac{\partial}{\partial w},$

where $\lambda\in\mathbb{C}^{*}$. Such a foliation has a non-degenerate
linearized singularity at $0$. Let $T$ be a harmonic current directed by
$\mathscr{F}$ which does not give mass to any of the two separatrices $(z=0)$
and $(w=0)$ and whose the trivial extension $\tilde{T}$ across $0$ is
$dd^{c}$-closed. The Lelong number of $T$ at $0$ describes the mass
distribution on the foliated space. In 2014 Nguyen proved that when
$\lambda\notin\mathbb{R}$, i.e. $0$ is a hyperbolic singularity, the Lelong
number at $0$ vanishes. For the non-hyperbolic case $\lambda\in\mathbb{R}^{*}$
the article proves the following results. The Lelong number at $0$:

1) is strictly positive if $\lambda>0$;

2) vanishes if $\lambda\in\mathbb{Q}_{<0}$;

3) vanishes if $\lambda<0$ and $T$ is invariant under the action of some
cofinite subgroup of the monodromy group.

Keywords: Holomorphic foliation, Harmonic current, Non-hyperbolic linearized
singularity, Lelong number.

##  1 Introduction

The dynamical properties of holomorphic foliations have drawn great attention
recently [40]. Let us see one of the recent interesting results.

######  Theorem 1.1 (Dinh-Nguyên-Sibony [20]).

Let $\mathscr{F}$ be a holomorphic foliation with only hyperbolic
singularities in a compact Kähler surface $(X,\omega)$. Assume that
$\mathscr{F}$ admits no directed positive closed current. Then there exists a
unique positive $dd^{c}$-closed current $T$ of mass $1$ directed by
$\mathscr{F}$.

The first version was stated for $X=\mathbb{P}^{2}$ and proved by Fornæss-
Sibony [28]. Later Dinh-Sibony proved unique ergodicity for foliations in
$\mathbb{P}^{2}$ with an invariant curve [23]. So one may expect to describe
recurrence properties of leaves by studying the density distribution of
directed harmonic currents. One has the following the result about leaves.

######  Theorem 1.2 (Fornæss-Sibony [28]).

Let $(X,\mathscr{F},E)$ be a hyperbolic foliation on a compact complex surface
$X$ with singular set $E$. Assume that

  1. 1.

there is no invariant analytic curve;

  2. 2.

all the singularities are hyperbolic;

  3. 3.

there is no non-constant holomorphic map $\mathbb{C}\rightarrow X$ such that
out of $E$ the image of $\mathbb{C}$ is locally contained in a leaf.

Then every harmonic current $T$ directed by $\mathscr{F}$ gives no mass to
each single leaf.

A practical way to measure the density of harmonic currents is to use the
notion of Lelong number introduced by Skoda [49]. Indeed Theorem 1.2 above is
equivalent to the statement that the Lelong number of $T$ vanishes everywhere
outside $E$. Another result holds near hyperbolic singularities.

######  Theorem 1.3 (Nguyên [34]).

Let $(\mathbb{D}^{2},\mathscr{F},\\{0\\})$ be a holomorphic foliation on the
unit bidisc $\mathbb{D}^{2}$ defined by the linear vector field
$Z(z,w)=z\,\frac{\partial}{\partial z}+\lambda\,w\,\frac{\partial}{\partial
w},$ where $\lambda\in\mathbb{C}\backslash\mathbb{R}$, that is to say, $0$ is
a hyperbolic singularity. Let $T$ be a harmonic current directed by
$\mathscr{F}$ which does not give mass to any of the two separatrices $(z=0)$
and $(w=0)$. Then the Lelong number of $T$ at $0$ vanishes.

Nguyên proved that the Lelong number of any directed harmonic current which
gives no mass to invariant hyperplanes, vanishes near weakly hyperbolic
singularities in $\mathbb{C}^{n}$ [40]. This result is optimal, see [25]. The
mass-distribution problem would be completed once the behaviour of harmonic
currents near non-hyperbolic and near degenerate singularities would be
understood.

The present paper answers (partly) the problem in the non-hyperbolic
linearizable singularity case.

######  Theorem 1.4.

Let $(\mathbb{D}^{2},\mathscr{F},\\{0\\})$ be a holomorphic foliation on the
unit bidisc $\mathbb{D}^{2}$ defined by the linear vector field
$Z(z,w)=z\,\frac{\partial}{\partial z}+\lambda\,w\,\frac{\partial}{\partial
w}$, where $\lambda\in\mathbb{R}^{*}$. Let $\mathscr{F}$ be a harmonic current
directed by $\mathscr{F}$ which does not give mass to any of the two
separatrices $(z=0)$ and $(w=0)$. Then the Lelong number of $T$ at $0$

  * •

is strictly positive if $\lambda>0$,

  * •

vanishes if $\lambda\in\mathbb{Q}_{<0}$.

For the concerned foliation $(\mathbb{D}^{2},\mathscr{F},\\{0\\})$, a local
leaf $P_{\alpha}$, with $\alpha\in\mathbb{C}^{*}$, can be parametrized by
$(z,w)=(e^{-v+iu},\alpha\,e^{-\lambda v+i\lambda u})$. The monodromy group
around the singularity is generated by $(z,w)\mapsto(z,e^{2\pi i\lambda}w)$.
It is a cyclic group of finite order when $\lambda\in\mathbb{Q}^{*}$, of
infinite order when $\lambda\notin\mathbb{Q}$.

It is now ready to introduce the notion of periodic current, an essential tool
of this paper. A directed harmonic current $T$ is called periodic if it is
invariant under some cofinite subgroup of the monodromy group, i.e. under the
action of $(z,w)\mapsto(z,e^{2k\pi i\lambda}w)$ for some
$k\in\mathbb{Z}_{>0}$. Observe that if $\lambda\in\mathbb{Q}^{*}$ then any
directed harmonic current is periodic. But when $\lambda\notin\mathbb{Q}^{*}$,
the periodicity is a nontrivial assumption.

######  Theorem 1.5.

Using the same notation as above, the Lelong number of $T$ at the singularity
is 0 when $\lambda<0$ and the current is periodic, in particular, when
$\lambda\in\mathbb{Q}_{<0}$.

It remains open to determine the possible Lelong number values of non-periodic
$T$ when $\lambda<0$ is irrational.

##  2 Background

To start with, recall the definition of singular holomorphic foliation on a
complex surface $M$.

######  Definition 2.1.

Let $E\subset M$ be some closed subset, possibly empty, such that
$\overline{M\backslash E}=M$. A singular holomorphic foliation
$(M,E,\mathscr{F})$ consists of a holomorphic atlas
$\\{(\mathbb{U}_{i},\Phi_{i})\\}_{i\in I}$ on $M\backslash E$ which satisfies
the following conditions.

  1. (1)

For each $i\in I$,
$\Phi_{i}:\mathbb{U}_{i}\rightarrow\mathbb{B}_{i}\times\mathbb{T}_{i}$ is a
biholomorphism, where $\mathbb{B}_{i}$ and $\mathbb{T}_{i}$ are domains in
$\mathbb{C}$.

  2. (2)

For each pair $(\mathbb{U}_{i},\Phi_{i})$ and $(\mathbb{U}_{j},\Phi_{j})$ with
$\mathbb{U}_{i}\cap\mathbb{U}_{j}\neq\emptyset$, the transition map

$\Phi_{ij}:=\Phi_{i}\circ\Phi_{j}^{-1}:\Phi_{j}(\mathbb{U}_{i}\cap\mathbb{U}_{j})\rightarrow\Phi_{i}(\mathbb{U}_{i}\cap\mathbb{U}_{j})$

has the form

$\Phi_{ij}(b,t)=\big{(}\Omega(b,t),\Lambda(t)\big{)},$

where $(b,t)$ are the coordinates on $\mathbb{B}_{j}\times\mathbb{T}_{j}$, and
the functions $\Omega$, $\Lambda$ are holomorphic, with $\Lambda$ independent
of $b$.

Each open set $\mathbb{U}_{i}$ is called a flow box. For each
$c\in\mathbb{T}_{i}$, the Riemann surface $\Phi_{i}^{-1}\\{t=c\\}$ in
$\mathbb{U}_{i}$ is called a plaque. The property (2) above insures that in
the intersection of two flow boxes, plaques are mapped to plaques.

A leaf $L$ is a minimal connected subset of $M$ such that if $L$ intersects a
plaque, it contains that plaque. A transversal is a Riemann surface immersed
in $M$ which is transverse to each leaf of $M$.

The local theory of singular holomorphic foliations is closely related to
holomorphic vector fields. One recalls some basic concepts in $\mathbb{C}^{2}$
([40], [14]).

######  Definition 2.2.

Let $Z=P(z,w)\frac{\partial}{\partial z}+Q(z,w)\frac{\partial}{\partial w}$ be
a holomorphic vector field defined in a neighborhood $\mathbb{U}$ of
$(0,0)\in\mathbb{C}^{2}$. One says that $Z$ is

  1. (1)

singular at $(0,0)$ if $P(0,0)=Q(0,0)=0$,

  2. (2)

linear if it can be written as

$Z=\lambda_{1}z\frac{\partial}{\partial
z}+\lambda_{2}w\frac{\partial}{\partial w}$

where $\lambda_{1}$, $\lambda_{2}\in\mathbb{C}$ are not simultaneously zero.

  3. (3)

linearizable if it is linear after a biholomorphic change of coordinates.

Suppose the holomorphic vector field $Z=P\frac{\partial}{\partial
z}+Q\frac{\partial}{\partial w}$ admits a singularity at the origin. Let
$\lambda_{1}$, $\lambda_{2}$ be the eigenvalues of the Jacobian matrix
$\textstyle\left(\\!\begin{array}[]{cc}P_{z}&P_{w}\\\
Q_{z}&Q_{w}\end{array}\\!\right)$ at the origin.

######  Definition 2.3.

The singularity is non-degenerate if both $\lambda_{1}$, $\lambda_{2}$ are
non-zero. This condition is biholomorphically invariant.

In this article, all singularities are assumed to be non-degenerate. Then the
foliation defined by integral curves of $Z$ has an isolated singularity at
$0$. Degenerate singularities are studied in [14]. Seidenberg’s reduction
theorem [46] shows that degenerate singularites can be resolved into non-
degenerate ones after finitely many blow-ups.

######  Definition 2.4.

A singularity of $Z$ is hyperbolic if the quotient
$\lambda:=\frac{\lambda_{1}}{\lambda_{2}}\in\mathbb{C}^{*}\backslash\mathbb{R}$.
It is non-hyperbolic if $\lambda\in\mathbb{R}^{*}$. It is in the Poincaré
domain if $\lambda\notin\mathbb{R}_{\leqslant 0}$. It is in the Siegel domain
if $\lambda\in\mathbb{R}_{<0}$.

One can verify that the quotient is unchanged by multiplication of $Z$ by any
non-vanishing holomorphic function.

One could consider $\lambda^{-1}=\frac{\lambda_{2}}{\lambda_{1}}$ instead of
$\lambda$, but then $\lambda\notin\mathbb{R}$ iff
$\lambda^{-1}\notin\mathbb{R}$. Thus the notion of hyperbolicity is well-
defined. Also, being non-hyperbolic, in Poincaré domain or in Siegel domain,
is well-defined. The complex number $\lambda$ will be called eigenvalue of $Z$
at the singularity, with an inessential abuse due to this exchange
$\lambda\leftrightarrow\lambda^{-1}$. The unordered pair
$\\{\lambda,\lambda^{-1}\\}$ is invariant under local biholomorphic changes of
coordinates.

Consider a holomorphic foliation $(M,E,\mathscr{F})$ where $E$ is discrete.
When one tries to linearize a vector field near an isolated singularity, one
has to divide power series coefficients by quantities
$\lambda_{1}\,m_{1}+\lambda_{2}\,m_{2}-\lambda_{j}$ for $j=1,2$ where $m_{1}$,
$m_{2}\in\mathbb{Z}_{\geqslant 1}$. To ensure convergence, these quantities
have to be nonzero and not too close to zero.

The resonances of $(\lambda_{1},\lambda_{2})\in\mathbb{C}^{2}$ are defined by

$\mathcal{R}:=\\{(m_{1},m_{2},j)\in\mathbb{Z}^{3}~{}|~{}m_{1},\,m_{2}\in\mathbb{Z}_{\geqslant
1},~{}\lambda_{1}\,m_{1}+\lambda_{2}\,m_{2}-\lambda_{j}=0\\}.$

Notice that the set
$\\{\lambda_{1}\,m_{1}+\lambda_{2}\,m_{2}-\lambda_{j}~{}|~{}m_{1},m_{2}\in\mathbb{Z}_{\geqslant
1}\\}$ has zero as a limit point if and only if the singularity is in the
Siegel domain.

We are now ready to state some linearization results in $\mathbb{C}^{2}$.

######  Theorem 2.5 (Poincaré [5]).

A singular holomorphic vector field with a non-resonant linear part, i.e.
$\mathcal{R}$ is empty, such that the eigenvalue $\lambda$ is in the Poincaré
domain, is holomorphically equivalent to its linear part.

That is to say, a singularity is linearizable if
$\lambda\notin\mathbb{R}_{\leqslant 0}\cup\mathbb{Q}$. To get linearization
for $\lambda$ in the Siegel domain, the following result assumes the more
advanced Brjuno condition.

######  Theorem 2.6 (Brjuno [5, 11]).

A singular holomorphic vector field with a non-resonant linear part is
holomorphically linearizable if its eigenvalue $\lambda\in\mathbb{R}$
satisfies the condition

$\sum_{n\geqslant 1}\frac{\log q_{n+1}}{q_{n}}<\infty,$

where $p_{n}/q_{n}$ is the $n^{\sl th}$ approximant of the continuous fraction
expansion of $\lambda$.

In this article, all singularities are assumed to be linearizable. Let
$(\mathbb{D}^{2},\mathscr{F},\\{0\\})$ be a holomorphic foliation on the unit
bidisc $\mathbb{D}^{2}$ defined by the linear vector field
$Z=z\frac{\partial}{\partial z}+\lambda w\frac{\partial}{\partial w}$ with
$\lambda\in\mathbb{R}^{*}$. One may assume $0<|\lambda|\leqslant 1$ after
switching $z$ and $w$ if necessary. There are always two separatrices
$\\{z=0\\}$ and $\\{w=0\\}$. Other leaves can be parametrized as

$L_{\alpha}:=\\{(z,w)=\psi_{\alpha}(\zeta):=(e^{i\,\zeta},\alpha\,e^{i\,\lambda\,\zeta})=(e^{-v+i\,u},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})\\}\quad{\scriptstyle{(\alpha\neq
0)}},$

where $\zeta=u+iv\in\mathbb{C}$. The map

$\displaystyle\Psi:\mathbb{C}\times\mathbb{C}^{*}$
$\displaystyle\longrightarrow\mathbb{C}^{2},$ $\displaystyle(\zeta,\alpha)$
$\displaystyle\longmapsto(e^{i\,\zeta},\alpha\,e^{i\,\lambda\,\zeta}),$

is locally biholomorphic. Here $\alpha$ is the coordinate on the transversal
and $\zeta$ is the coordinate on leaves. It is not injective since
$\Psi(\zeta+2\pi,\alpha)=\Psi(\zeta,\alpha\,e^{2\pi i\lambda})$.

Two numbers $\alpha$, $\beta\in\mathbb{C}^{*}$ are equivalent
$\alpha\sim\beta$ if $\beta=e^{2k\pi i\lambda}\alpha$ for some
$k\in\mathbb{Z}$. The following statements are equivalent:

  * •

$\alpha\sim\beta$;

  * •

$L_{\alpha}=L_{\beta}$;

  * •

$\psi_{\alpha}=\psi_{\beta}\circ(\text{translation of }2k\pi)$ for some
$k\in\mathbb{Z}$.

Let $\mathscr{C}_{\mathscr{F}}$ (resp. $\mathscr{C}_{\mathscr{F}}^{1,1}$)
denote the space of functions (resp. forms of bidegree (1,1)) defined on
leaves of the foliation which are compactly supported on $M\backslash E$,
leafwise smooth and transversally continuous. A form
$\iota\in\mathscr{C}_{\mathscr{F}}^{1,1}$ is said to be positive if its
restriction to every plaque is a positive (1,1)-form.

A directed harmonic current $T$ on $\mathscr{F}$ is a continuous linear form
on $\mathscr{C}_{\mathscr{F}}^{1,1}$ satisfying the following two conditions:

  1. 1.

$i\partial\bar{\partial}T=0$ in the weak sense, i.e.
$T(i\partial\bar{\partial}f)=0$ for all $f\in\mathscr{C}_{\mathscr{F}}$, where
in the expression $i\partial\bar{\partial f}$, one only considers
$\partial\bar{\partial}$ along the leaves;

  2. 2.

$T$ is positive, that is $T(\iota)\geqslant 0$ for all positive forms
$\iota\in\mathscr{C}_{\mathscr{F}}^{1,1}$.

According to [8], a directed harmonic current $T$ on a flow box
$\mathbb{U}\cong\mathbb{B}\times\mathbb{T}$ can be locally expressed as

$T=\int_{\alpha\in\mathbb{T}}h_{\alpha}[P_{\alpha}]d\mu(\alpha).$ (1)

The $h_{\alpha}$ are non-negative harmonic functions on the local leaves
$P_{\alpha}$ and $\mu$ is a Borel measure on the transversal $\mathbb{T}$. If
$h_{\alpha}=0$ at some point on $P_{\alpha}$, then by the mean value theorem
$h_{\alpha}\equiv 0$. For all such $\alpha\in\mathbb{T}$, we replace
$h_{\alpha}$ by the constant function $1$ and we set $d\mu(\alpha)=0$. Thus we
get a new expression of $T$ where $h_{\alpha}>0$ for all
$\alpha\in\mathbb{T}$.

Such an expression is not unique since
$T=\int_{\alpha\in\mathbb{T}}\big{(}h_{\alpha}\,g(\alpha)\big{)}[P_{\alpha}]\big{(}\frac{1}{g(\alpha)}\,d\mu(\alpha)\big{)}$
for any bounded positive function $g:\mathbb{T}\rightarrow\mathbb{R}_{>0}$.
The expression is unique after normalization, which means that for each
$\alpha\in\mathbb{T}$ one fixes $h_{\alpha}(z_{0},w_{0})=1$ at some point
$(z_{0},w_{0})\in P_{\alpha}$.

Each harmonic function $h_{\alpha}$ on the leaf $V_{\alpha}$ can be pulled
back by the parametrization $\Psi$ as the harmonic function

$H_{\alpha}(u,v):=h_{\alpha}\big{(}e^{-v+iu},\alpha\,e^{-\lambda v+i\lambda
u}\big{)}.$

The domain of definition for $u$, $v$ will be precisely described later in
this section.

In Section 1 the notion of periodic current was introduced. Here is an
equivalent characterization.

######  Proposition 2.7.

A directed harmonic current $T$ is periodic if and only if there exists some
$k\in\mathbb{Z}_{>0}$ such that $H_{\alpha}(u+2k\pi,v)=H_{\alpha}(u,v)$ for
all $u,v$ and for $\mu$-almost all $\alpha$.

###### Proof.

By definition $T$ is invariant under $(z,w)\mapsto(z,e^{2k\pi i\lambda}w)$ for
some $k\in\mathbb{Z}_{>0}$, which is equivalent to
$H_{\alpha}(u+2k\pi,v)=H_{\alpha}(u,v)$ for all $u,v$ and $\mu$-almost all
$\alpha$. ∎

A current $T$ of the form (1) is $dd^{c}$-closed on
$\mathbb{D}^{2}\backslash\\{0\\}$. But its trivial extension $\tilde{T}$
across the singularity $0$ is not necessarily $dd^{c}$-closed on
$\mathbb{D}^{2}$. It is true when $T$ is compactly supported, for example when
$T$ is a localization of a current on a compact manifold, by the following
argument of Dinh-Nguyên-Sibony in [17] Lemma 2.5.

Let $T$ be a directed harmonic current on $M\backslash E$, where $M$ is a
compact complex manifold and $E$ is a finite set. The current $T$ can be
extended by $0$ as a positive current $\tilde{T}$ on $M$. Next, apply

######  Theorem 2.8 (Alessandrini-Bassanelli [2]).

Let $\Omega$ be an open subset of $\mathbb{C}^{n}$ and $Y$ an analytic subset
of $\Omega$ of dimension $<p$. Suppose $T$ is a negative current of
bidimension $(p,p)$ on $\Omega\backslash Y$ such that $dd^{c}T\geqslant 0$.
Suppose $\tilde{T}$ is the trivial extension of $T$ across $Y$ by $0$. Then
$dd^{c}\tilde{T}\geqslant 0$ on $\Omega$.

Here $-T$ is a negative current of bidimension $(1,1)$ on $M\backslash E$ with
$dd^{c}(-T)\geqslant 0$ and $E$ has dimension $0$. So for the trivial
extension $\tilde{T}$ on $M$ one has $dd^{c}(-\tilde{T})\geqslant 0$. Moreover
$\tilde{T}$ is compactly supported since $M$ is compact. Thus

$\left<dd^{c}\tilde{T},1\right>=\left<\tilde{T},dd^{c}1\right>=0.$

Combining with $dd^{c}\tilde{T}\leqslant 0$ from the extension theorem, one
concludes that $dd^{c}\tilde{T}=0$ on $M$. Thus locally near any singularity,
the trivial extension $\tilde{T}$ is $dd^{c}$-closed.

Let $\beta:=idz\wedge d\bar{z}+idw\wedge d\bar{w}$ be the standard Kähler
form. The mass of $T$ on a domain $U\subset\mathbb{D}^{2}$ is denoted by
$|\\!|T|\\!|_{U}:=\int_{U}T\wedge\beta$. In this article, all currents are
assumed to have finite mass on $\mathbb{D}^{2}$.

######  Definition 2.9.

(See [37, Subsection 2.4]). Let $T$ be a directed harmonic current on
$(\mathbb{D}^{2},\mathscr{F},\\{0\\})$. We define the Lelong number by the
limit

$\mathscr{L}(T,0)=\limsup\limits_{r\rightarrow 0+}\frac{1}{\pi
r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}\in[0,+\infty].$

The limit can be infinite when the trivial extension $\tilde{T}$ across the
origin is not $dd^{c}$-closed [37, Example 2.11]. When $\tilde{T}$ is
$dd^{c}$-closed, the following theorem assures the finiteness.

######  Theorem 2.10 (Skoda [49]).

Let $T$ be a positive $dd^{c}$-closed $(1,1)$-current in $\mathbb{D}^{2}$.
Then the function $r\mapsto\frac{1}{\pi r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}$
is increasing with $r\in(0,1]$.

In our case, the function

$r\mapsto\frac{1}{\pi r^{2}}|\\!|\tilde{T}|\\!|_{r\mathbb{D}^{2}}=\frac{1}{\pi
r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}},$

is increasing with $r\in(0,1]$. In particular

$\mathscr{L}(T,0)=\lim\limits_{r\rightarrow 0+}\frac{1}{\pi
r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}\in[0,\tfrac{1}{\pi}|\\!|T|\\!|_{\mathbb{D}^{2}}].$

To calculate $|\\!|T|\\!|_{\mathbb{D}^{2}}$ and $\mathscr{L}(T,0)$ one shall
study $\Psi^{-1}(r\,\mathbb{D}^{2})$ for $r\in(0,1]$. Define
$P_{\alpha}:=L_{\alpha}\cap\mathbb{D}^{2}$ and
$P_{\alpha}^{(r)}:=L_{\alpha}\cap r\,\mathbb{D}^{2}$. Define
$\log^{+}(x):=\max\\{0,\log(x)\\}$ for $x>0$.

######  Lemma 2.11.

The range of $(u,v)$ for a point $(z,w)\in P_{\alpha}$ and $P_{\alpha}^{(r)}$
is either an upper-half plane or a horizontal strip. More precisely,

  1. 1.

when $\lambda>0$,

$\displaystyle(z,w)\in P_{\alpha}$ $\displaystyle\Longleftrightarrow
v>\frac{\log^{+}|\alpha|}{\lambda},$ $\displaystyle(z,w)\in P_{\alpha}^{(r)}$
$\displaystyle\Longleftrightarrow\left\\{\begin{aligned}
&v>\frac{\log|\alpha|-\log r}{\lambda}&\quad{\scriptstyle{(|\alpha|\geqslant
r^{1-\lambda})}},\\\ &v>-\log
r&\quad{\scriptstyle{(|\alpha|<r^{1-\lambda})}}.\end{aligned}\right.$

  2. 2.

when $\lambda<0$, $P_{\alpha}=\emptyset$ for $|\alpha|\geqslant 1$,
$P_{\alpha}^{(r)}=\emptyset$ for $|\alpha|\geqslant r^{1-\lambda}$ and for the
other $\alpha$

$\displaystyle(z,w)\in P_{\alpha}$ $\displaystyle\Longleftrightarrow
0<v<\frac{\log|\alpha|}{\lambda},$ $\displaystyle(z,w)\in P_{\alpha}^{(r)}$
$\displaystyle\Longleftrightarrow-\log r<v<\frac{\log|\alpha|-\log
r}{\lambda}.$

###### Proof.

Recall that $(z,w)=(e^{-v+i\,u},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})$ on
$L_{\alpha}$. So for any $r\in(0,1]$, $(z,w)\in P_{\alpha}^{(r)}$ if and only
if both $|z|=e^{-v}<r$ and $|w|=|\alpha|\,e^{-\lambda\,v}<r$.

When $\lambda>0$ one has $v>-\log r$ and $v>\frac{\log|\alpha|-\log
r}{\lambda}$. In particular for $r=1$, one has $v>0$ and
$v>\frac{\log|\alpha|}{\lambda}$.

When $\lambda<0$ one has $-\log r<v<\frac{\log|\alpha|-\log r}{\lambda}$. In
particular for $r=1$, one has $0<v<\frac{\log|\alpha|}{\lambda}$. If there is
no solution for $v$ then $P_{\alpha}^{(r)}=\emptyset$.∎

When $\lambda>0$, the range of $v$ is unbounded for each fixed
$\alpha\in\mathbb{C}^{*}$.

Figure 1: The region of $(|\alpha|,v)$ for $P_{\alpha}$ Figure 2: The region
of $(|\alpha|,v)$ for $P_{\alpha}^{(r)}$

When $\lambda<0$, the range of $v$ is bounded for each fixed $\alpha$.

Figure 3: The region of $(|\alpha|,v)$ for $P_{\alpha}$ Figure 4: The region
of $(|\alpha|,v)$ for $P_{\alpha}^{(r)}$

In this article, the notations $\lesssim$ and $\gtrsim$ stand for inequalities
up to a multiplicative non-zero constant depending only on $\lambda$. We write
$\approx$ when both inequalities are satisfied.

##  3 Geometry of leaves when $\lambda>0$

For any $\alpha\in\mathbb{C}^{*}$ fixed, the leaf $L_{\alpha}$ is contained in
a real 3-dimensional Levi flat CR manifold $|w|=|\alpha|\,|z|^{\lambda}$,
which can be viewed as a curve in $|z|=e^{-v}$, $|w|=|\alpha|\,e^{-\lambda v}$
coordinates. The norms $|z|$ and $|w|$ depends only on $v$. When
$v\rightarrow+\infty$, the point on the leaf tends to the singularity $(0,0)$
described by the following figures:

Figure 5: Case $|\alpha|<1$ Figure 6: Case $|\alpha|\geqslant 1$

If one fixes some $v=-\log r$, then $|z|=r$ and $|w|=|\alpha|\,r^{\lambda}$ is
fixed. The set
$\mathbb{T}^{2}_{r}:=\\{(z,w)\in\mathbb{D}^{2}~{}:~{}|z|=r,|w|=|\alpha|\,r^{\lambda}\\}$
is a torus and the intersection of the leaf $L_{\alpha}$ with this torus is a
smooth curve $L_{\alpha,r}:=L_{\alpha}\cap\mathbb{T}^{2}_{r}$.

When $\lambda\in\mathbb{Q}$, this curve $L_{\alpha,r}$ is closed.

Figure 7: a closed curve on a torus

When $\lambda\notin\mathbb{Q}$, this curve $L_{\alpha,r}$ is dense on the
torus $\mathbb{T}_{r}^{2}$.

Figure 8: 2 loops Figure 9: 20 loops

In this case the two curves $L_{\alpha,r}$ and $L_{\alpha\,e^{2\pi
i\lambda},r}$ are two different parametrizations of the same image. The green
curve in Figure 8 is not only the image of $L_{\alpha,r}$ for
$u\in[2\pi,4\pi)$ but also the image of $L_{\alpha\,e^{2\pi i\lambda},r}$ for
$u\in[0,2\pi)$. This raises ambiguity while doing normalization on a leaf
$L_{\alpha}$.

Such ambiguity can be resolved once one restricts everything to an open subset
$U_{\epsilon}:=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}{\rm
arg}(z)\in(0,2\pi-\epsilon),z\neq 0,w\neq 0\\}$ for some fixed
$\epsilon\in[0,\pi)$. Any leaf $L_{\alpha}$ on $U_{\epsilon}$ decomposes into
a disjoint union of infinitely many components:

$L_{\alpha}\cap
U_{\epsilon}=\bigcup\limits_{k\in\mathbb{Z}}\big{\\{}(e^{-v+iu},\alpha\,e^{2k\pi
i\lambda}\,e^{-\lambda v+i\lambda
u})~{}|~{}u\in(0,2\pi-\epsilon),v>\textstyle\frac{\log^{+}|\alpha|}{\lambda}\big{\\}}.$

Such a parametrization is yet not unique. For example for any
$k_{0}\in\mathbb{Z}$ one can do

$L_{\alpha}\cap
U_{\epsilon}=\bigcup\limits_{k\in\mathbb{Z}}\big{\\{}(e^{-v+iu},\alpha\,e^{2k\pi
i\lambda}\,e^{-\lambda v+i\lambda
u})~{}|~{}u\in(2k_{0}\pi,2k_{0}\pi+2\pi-\epsilon),v>\textstyle\frac{\log^{+}|\alpha|}{\lambda}\big{\\}}.$

The parametrization is unique once one fixes $k_{0}$, for example $k_{0}=0$.
But I leave a hint that all other choices of $k_{0}$ will be useful in a later
proof.

Once one specifies the range of $u$ by fixing $k_{0}=0$, then the leaf on
$U_{\epsilon}$ is uniquely parametrized.

Figure 10: The curve $L_{1,1}$ in red and the curve $L_{e^{2\pi i\lambda},1}$
in green

Later the limit case when $\epsilon=0$,
$U:=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}{\rm arg}(z)\in(0,2\pi),z\neq 0,w\neq
0\\}=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}z\notin\mathbb{R}_{\geqslant 0},w\neq
0\\}$ will be used.

##  4 Rational case: $\lambda=\frac{a}{b}\in\mathbb{Q}$, $\lambda\in(0,1]$

Say $\lambda=\frac{a}{b}$ where $a,b\in\mathbb{Z}_{\geqslant 1}$ are coprime.
Then in $\mathbb{D}^{2}$, for any $\alpha\in\mathbb{C}^{*}$, the union
$L_{\alpha}\cup\\{0\\}$ is the algebraic curve
$\\{w^{b}=\alpha^{b}\,z^{a}\\}\cap\mathbb{D}^{2}$. In other words, every leaf
is a separatix. In this section it will be shown that any directed harmonic
current $T$ has non-zero Lelong number.

The parametrization map
$\psi_{\alpha}(\zeta):=(e^{i\zeta},\alpha\,e^{i\lambda\zeta})$ is now
periodic: $\psi_{\alpha}(\zeta+2\pi b)=\psi_{\alpha}(\zeta)$. Let $T$ be a
directed harmonic current. Then $T|_{P_{\alpha}}$ has the form
$h_{\alpha}(z,w)[P_{\alpha}]$. Let

$H_{\alpha}(u+iv):=h_{\alpha}\circ\psi_{\alpha}\big{(}u+iv+i\frac{\log^{+}|\alpha|}{\lambda}\big{)}.$

It is a positive harmonic function for $\mu$-almost all
$\alpha\in\mathbb{C}^{*}$ defined in a neighborhood of the upper half plane
$\mathbb{H}:=\\{(u+iv)\in\mathbb{C}~{}|~{}v>0\\}$. Moreover it is periodic:
$H_{\alpha}(u+iv)=H_{\alpha}(u+2\pi b+iv)$. Periodic harmonic functions can be
characterized by

######  Lemma 4.1.

Let $F(u,v)$ be a harmonic function in a neighborhood of $\mathbb{H}$. If
$F(u,v)=F(u+2\pi b,v)$ for all $(u,v)\in\mathbb{H}$, then

$F(u,v)=\sum\limits_{k\in\mathbb{Z},k\neq
0}\big{(}a_{k}\,e^{\frac{kv}{b}}\cos(\textstyle\frac{ku}{b})+b_{k}\,e^{\frac{kv}{b}}\sin(\frac{ku}{b})\big{)}+a_{0}+b_{0}\,v,$

for some $a_{k}$, $b_{k}\in\mathbb{R}$. Moreover, if $F|_{\mathbb{H}}\geqslant
0$, then $a_{0},b_{0}\geqslant 0$.

###### Proof.

By periodicity

$F(u,v)=\sum\limits_{k=1}^{\infty}\big{(}A_{k}(v)\,\cos(\textstyle\frac{ku}{b})+B_{k}(v)\,\sin(\frac{ku}{b})\big{)}+A_{0}(v),$

for some functions $A_{k}(v)$, $B_{k}(v)$. They are smooth since $F$ is
harmonic. Moreover

$0=\Delta
F(u,v)=\sum\limits_{k=1}^{\infty}\Big{(}\big{(}A_{k}^{\prime\prime}(v)-(\textstyle\frac{k}{b})^{2}\,A_{k}(v)\big{)}\,\cos(\frac{ku}{b})+\big{(}B_{k}^{\prime\prime}(v)-(\frac{k}{b})^{2}\,B_{k}(v)\big{)}\,\sin(\frac{ku}{b})\Big{)}+A_{0}^{\prime\prime}(v).$

Thus

$A_{k}^{\prime\prime}(v)=(\textstyle\frac{k}{b})^{2}\,A_{k}(v),\ \ \ \
B_{k}^{\prime\prime}(v)=(\frac{k}{b})^{2}\,B_{k}(v),\ \ \ \
A_{0}^{\prime\prime}(v)=0.$

Hence

$A_{k}(v)=a_{k}\,e^{\frac{kv}{b}}+a_{-k}\,e^{-\frac{kv}{b}},\ \ \ \
B_{k}(v)=b_{k}\,e^{\frac{kv}{b}}-b_{-k}\,e^{-\frac{kv}{b}},\ \ \ \
A_{0}(v)=a_{0}+b_{0}\,v,$

for some $a_{k}$, $a_{-k}$, $b_{k}$, $b_{-k}\in\mathbb{R}$. One obtains the
equality.

If $F|_{\mathbb{H}}\geqslant 0$, then for any $v\geqslant 0$,

$\int_{u=0}^{2\pi b}F(u,v)du=2\pi b(a_{0}+b_{0}\,v)\geqslant 0.$

Thus $a_{0},b_{0}\geqslant 0.$ ∎

For $\alpha,\beta\in\mathbb{C}^{*}$, the two maps $\psi_{\alpha}$ and
$\psi_{\beta}$ parametrize the same leaf $L_{\alpha}=L_{\beta}$ if and only if
$\beta=\alpha\,e^{2\pi i\frac{k}{b}}$ for some $k\in\mathbb{Z}$, i.e. $\alpha$
and $\beta$ differ from multiplying a $b^{\sl th}$-root of unity. Thus a
transversal can be chosen as the section
$\mathbb{T}:=\\{\alpha\in\mathbb{C}^{*}~{}|~{}{\rm
arg}(\alpha)\in[0,\frac{2\pi}{b})\\}$. One fixes a normalization by setting
$H_{\alpha}(0)=h_{\alpha}\circ\psi_{\alpha}(i\frac{\log^{+}|\alpha|}{\lambda})=1$.

The mass of the current $T$ is

$|\\!\\!|T|\\!\\!|_{\mathbb{D}^{2}}=\int_{(z,w)\in\mathbb{D}^{2}}T\wedge
i\partial\bar{\partial}(|z|^{2}+|w|^{2}).$

In particular, one calculates the $(1,1)$-form
$i\partial\bar{\partial}(|z|^{2}+|w|^{2})$ on $L_{\alpha}$, where
$z=e^{-v+iu},w=\alpha\,e^{-\lambda v+i\lambda u}$, using

$\displaystyle dz$ $\displaystyle=ie^{-v+iu}du-e^{-v+iu}dv,$ $\displaystyle
d\bar{z}$ $\displaystyle=-ie^{-v-iu}du-e^{-v-iu}dv,$ $\displaystyle dw$
$\displaystyle=i\alpha\,\lambda\,e^{-\lambda\,v+i\lambda\,u}du-\alpha\,\lambda\,e^{-\lambda\,v+i\lambda\,u}dv,$
$\displaystyle d\bar{w}$
$\displaystyle=-i\bar{\alpha}\,\lambda\,e^{-\lambda\,v-i\lambda\,u}du-\bar{\alpha}\,\lambda\,e^{-\lambda\,v-i\lambda\,u}dv,$

whence

$\displaystyle i\partial\bar{\partial}(|z|^{2}+|w|^{2})$
$\displaystyle=idz\wedge d\bar{z}+idw\wedge d\bar{w}$
$\displaystyle=2\big{(}e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v}\big{)}du\wedge
dv.$

Thus

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\int_{\alpha\in\mathbb{T}}h_{\alpha}(z,w)\int_{P_{\alpha}}i\partial\bar{\partial}(|z|^{2}+|w|^{2})\,d\mu(\alpha)$
$\displaystyle=\int_{\alpha\in\mathbb{T}}\int_{u=0}^{2\pi
b}\int_{v>0}H_{\alpha}(u+iv)\,2\big{(}e^{-2(v+\frac{\log^{+}|\alpha|}{\lambda})}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,(v+\frac{\log^{+}|\alpha|}{\lambda})}\big{)}\,du\wedge
dv\,d\mu(\alpha)$
$\displaystyle=\int_{\alpha\in\mathbb{T},|\alpha|<1}\int_{u=0}^{2\pi
b}\int_{v>0}H_{\alpha}(u+iv)\,2\big{(}e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v}\big{)}\,du\wedge
dv\,d\mu(\alpha)$ $\displaystyle\ \ \
+\int_{\alpha\in\mathbb{T},|\alpha|\geqslant 1}\int_{u=0}^{2\pi
b}\int_{v>0}H_{\alpha}(u+iv)\,2\big{(}|\alpha|^{-\frac{2}{\lambda}}\,e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v}\big{)}du\wedge
dv\,d\mu(\alpha).$

By Lemma 4.1,

$H_{\alpha}(u+iv)=\sum\limits_{k\in\mathbb{Z},k\neq
0}\big{(}a_{k}(\alpha)\,e^{\frac{kv}{b}}\cos(\textstyle\frac{ku}{b})+b_{k}(\alpha)\,e^{\frac{kv}{b}}\sin(\frac{ku}{b})\big{)}+a_{0}(\alpha)+b_{0}(\alpha)\,v,$
(2)

where $a_{0}(\alpha)$, $b_{0}(\alpha)$ are positive for $\mu$-almost all
$\alpha$. Thus

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$ $\displaystyle=2\pi
b\Big{\\{}\int_{\alpha\in\mathbb{T},|\alpha|<1}\int_{v>0}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ +\int_{\alpha\in\mathbb{T},|\alpha|\geqslant
1}\int_{v>0}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2(|\alpha|^{-\frac{2}{\lambda}}\,e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)\Big{\\}}$
$\displaystyle=2\pi
b\,\Big{\\{}\int_{|\alpha|<1}a_{0}(\alpha)\,(1+|\alpha|^{2}\,\lambda)d\mu(\alpha)+\int_{|\alpha|\geqslant
1}a_{0}(\alpha)\,(|\alpha|^{-\frac{2}{\lambda}}+\lambda)d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \
+\int_{|\alpha|<1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{2}\big{)}d\mu(\alpha)+\int_{|\alpha|\geqslant
1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\big{)}d\mu(\alpha)\Big{\\}}$
$\displaystyle\approx\int_{\alpha\in\mathbb{C}^{*}}a_{0}(\alpha)\,d\mu(\alpha)+\int_{\alpha\in\mathbb{C}^{*}}b_{0}(\alpha)\,d\mu(\alpha).$

The Lelong number can now be calculated as follows

$\displaystyle\mathscr{L}(T,0)$ $\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}$
$\displaystyle=\lim\limits_{r\rightarrow 0+}\frac{1}{r^{2}}2\pi
b\,\Big{\\{}\int_{\alpha\in\mathbb{T},|\alpha|<r^{1-\lambda}}\int_{v>-\log
r}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\,\int_{\alpha\in\mathbb{T},r^{1-\lambda}\leqslant|\alpha|<1}\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\,\int_{\alpha\in\mathbb{T},|\alpha|\geqslant 1}\int_{v>\frac{-\log
r}{\lambda}}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(|\alpha|^{-\frac{2}{\lambda}}e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)\Big{\\}}$
$\displaystyle=\lim\limits_{r\rightarrow 0+}2\pi
b\,\Big{\\{}\int_{\alpha\in\mathbb{T},|\alpha|<r^{1-\lambda}}a_{0}(\alpha)\,(1+\lambda\,|\alpha|^{2}\,r^{2\lambda-2})\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{\alpha\in\mathbb{T},|\alpha|\geqslant
r^{1-\lambda}}a_{0}(\alpha)\,(|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}+\lambda)\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{\alpha\in\mathbb{T},|\alpha|<r^{1-\lambda}}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{2}\,r^{2\lambda-2}-\log
r-\lambda\,|\alpha|^{2}\,r^{2\lambda-2}\,\log r\big{)}\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{\alpha\in\mathbb{T},r^{1-\lambda}\leqslant|\alpha|<1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}-\log
r-|\alpha|^{-\frac{2}{\lambda}}\,\lambda^{-1}\,r^{2\lambda-2}\log r$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \
+\log|\alpha|+\lambda^{-1}\,|\alpha|^{-\frac{2}{\lambda}}\,\log|\alpha|\,r^{2\lambda-2}\big{)}\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{\alpha\in\mathbb{T},|\alpha|\geqslant
1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}-\log
r-\lambda^{-1}\,|\alpha|^{-\frac{2}{\lambda}}\,r^{2\lambda-2}\log
r\big{)}\,d\mu(\alpha)\Big{\\}}.$

First one analyzes the $a_{0}(\alpha)$ part. When $|\alpha|<r^{1-\lambda}$,

$\displaystyle
1<1+\lambda\,|\alpha|^{2}\,r^{2\lambda-2}<1+\lambda\,r^{2-2\lambda}\,r^{2\lambda-2}=1+\lambda,$
(3)

is uniformly bounded with respect to $\alpha$ and $r$. When $|\alpha|\geqslant
r^{1-\lambda}$

$\displaystyle\lambda<|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}+\lambda<1+\lambda,$
(4)

is also uniformly bounded with respect to $\alpha$ and $r$. Thus

$\mathscr{L}(T,0)\approx\underbrace{\int_{\alpha\in\mathbb{T}}a_{0}(\alpha)\,d\mu(\alpha)}_{\text{linear
part}}+\underbrace{\lim\limits_{r\rightarrow 0+}\big{(}\text{$b_{0}(\alpha)$
part}\big{)}}_{\text{with $v$ part}}.$

Next one analyses the $b_{0}(\alpha)$ part.

######  Lemma 4.2.

The Lelong number of $T$ at $0$ is finite only if $b_{0}(\alpha)=0$ for
$\mu$-almost all $\alpha\in\mathbb{T}$.

###### Proof.

Suppose not, i.e.
$\int_{\alpha\in\mathbb{T}}b_{0}(\alpha)\,d\mu(\alpha)=B_{0}>0$. Then

$\displaystyle\mathscr{L}(T,0)$
$\displaystyle\geqslant\lim\limits_{r\rightarrow 0+}2\pi
b\Big{\\{}\int_{\alpha\in\mathbb{T},|\alpha|<r^{1-\lambda}}b_{0}(\alpha)\,\big{(}-\log
r\big{)}\,d\mu(\alpha)+\int_{\alpha\in\mathbb{T},|\alpha|\geqslant
r^{1-\lambda}}b_{0}(\alpha)\,\big{(}-\log r\big{)}\,d\mu(\alpha)\Big{\\}}$
$\displaystyle=2\pi b\,B_{0}\,\lim\limits_{r\rightarrow 0+}(-\log r)=+\infty,$

would contradict the finiteness of the Lelong number stated in Theorem 2.10. ∎

Thus one may assume $b_{0}(\alpha)=0$ for $\mu$-almost all
$\alpha\in\mathbb{T}$. Then the Lelong number

$\mathscr{L}(T,0)\approx\int_{\alpha\in\mathbb{T}}a_{0}(\alpha)\,d\mu(\alpha)\approx|\\!|T|\\!|_{\mathbb{D}^{2}},$

is strictly positive.

##  5 Irrational case $\lambda\notin\mathbb{Q}$, $\lambda\in(0,1)$

Now $\\{z=0\\}$ and $\\{w=0\\}$ are the only 2 separatrices in
$\mathbb{D}^{2}$. For each fixed $\alpha\in\mathbb{C}^{*}$, the map
$\psi_{\alpha}(\zeta)=(e^{i\,\zeta},\alpha\,e^{i\,\lambda\,\zeta})$ is
injective since $\lambda\notin\mathbb{Q}$.

Any pair of equivalent numbers $\alpha\sim\beta$, $\beta=\alpha\,e^{2k\pi
i\lambda}$, may provide us with two different normalizations $H_{\alpha}$ and
$H_{\beta}$ on the same leaf $L_{\alpha}=L_{\beta}$. A major task is to find
formulas for the mass and the Lelong number independent by the choice of
normalization.

Let $T$ be a harmonic current directed by $\mathscr{F}$. Then
$T|_{P_{\alpha}}$ has the form $h_{\alpha}(z,w)[P_{\alpha}]$. One may assume
that $h_{\alpha}$ is nowhere 0 for every $\alpha$. Let

$H_{\alpha}(u+iv):=h_{\alpha}\circ\psi_{\alpha}\big{(}u+iv+i\frac{\log^{+}|\alpha|}{\lambda}\big{)}.$

It is a positive harmonic function for $\mu$-almost all
$\alpha\in\mathbb{C}^{*}$ defined in a neighborhood of the upper half plane
$\mathbb{H}=\\{(u+iv)\in\mathbb{C}~{}|~{}v>0\\}$, determined by the Poisson
integral formula

$H_{\alpha}(u+iv)=\frac{1}{\pi}\,\int_{y\in\mathbb{R}}H_{\alpha}(y)\,\frac{v}{v^{2}+(y-u)^{2}}\,dy+C_{\alpha}\,v.$

One can normalize $H_{\alpha}$ by setting $H_{\alpha}(0)=1$. But by doing so
one may normalize data over the same leaf for multiple times. The ambiguity is
described by

######  Proposition 5.1.

If $\beta=\alpha\,e^{2k\,\pi\,i\,\lambda}$ for some $k\in\mathbb{Z}$, then the
two normalized positive harmonic functions $H_{\alpha}$ and $H_{\beta}$
satisfy

$H_{\alpha}(u+iv)=H_{\alpha}(2k\pi)\,H_{\beta}(u-2k\pi+iv).$

In other words, they differ by a translation and a multiplication by a non-
zero constant.

###### Proof.

When $|\alpha|<1$, by definition

$H_{\alpha}(u+iv)=h_{\alpha}(e^{-v+iu},\alpha\,e^{-\lambda\,v+i\,\lambda\,u}),\
\ \ \ H_{\alpha}(0)=h_{\alpha}(1,\alpha).$

Thus the normalized harmonic function is

$H_{\alpha}(u+iv)=\frac{h_{\alpha}(e^{-v+iu},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})}{h_{\alpha}(1,\alpha)},$

and for the same reason

$H_{\beta}(u+iv)=\frac{h_{\beta}(e^{-v+iu},\beta\,e^{-\lambda\,v+i\,\lambda\,u})}{h_{\beta}(1,\beta)}.$

The two functions $h_{\alpha}$ and $h_{\beta}$ are the positive harmonic
coefficient of $T$ on the same leaf $L_{\alpha}=L_{\beta}$, hence they differ
up to multiplication by a positive constant $C>0$,

$\displaystyle h_{\alpha}(e^{-v+iu},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})$
$\displaystyle=C\cdot
h_{\beta}(e^{-v+iu},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})$
$\displaystyle=C\cdot
h_{\beta}(e^{-v+iu},\beta\,e^{-2k\,\pi\,i\,\lambda}\,e^{-\lambda\,v+i\,\lambda\,u})$
$\displaystyle=C\cdot
h_{\beta}(e^{-v+i(u-2k\,\pi)},\beta\,e^{-\lambda\,v+i\,\lambda\,(u-2k\,\pi)}).$

Thus

$\displaystyle H_{\alpha}(u+iv)$
$\displaystyle=\frac{h_{\alpha}(e^{-v+iu},\alpha\,e^{-\lambda\,v+i\,\lambda\,u})}{h_{\alpha}(1,\alpha)}=\frac{C\cdot
h_{\beta}(e^{-v+i(u-2k\,\pi)},\beta\,e^{-\lambda\,v+i\,\lambda\,(u-2k\,\pi)})}{C\cdot
h_{\beta}(1,\alpha)}$
$\displaystyle=\frac{h_{\beta}(e^{-v+i(u-2k\,\pi)},\beta\,e^{-\lambda\,v+i\,\lambda\,(u-2k\,\pi)})}{h_{\beta}(1,\beta)}\cdot\frac{h_{\beta}(1,\beta)}{h_{\beta}(1,\alpha)}$
$\displaystyle=H_{\beta}(u-2k\,\pi+iv)\cdot\frac{h_{\beta}(1,\beta)}{h_{\beta}(1,\alpha)}.$

When $u=2k\,\pi$ and $v=0$ one has
$H_{\alpha}(2k\,\pi)=\frac{h_{\beta}(1,\beta)}{h_{\beta}(1,\alpha)}$. Thus one
gets the equality. The proof for the case $|\alpha|>1$ is similar. ∎

Take the open subset
$U:=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}z\notin\mathbb{R}_{\geqslant 0},w\neq
0\\}$.

Figure 11: Domain $U$ in coordinates $(z,w)$ Figure 12: Domain $U$ in
coordinates $(u,v)$

Any leaf $L_{\alpha}$ in $U$ is a disjoint union of infinitely many
components:

$L_{\alpha}\cap U=\bigcup\limits_{k\in\mathbb{Z}}\tilde{L}_{\alpha\,e^{2k\pi
i\lambda}}:=\bigcup\limits_{k\in\mathbb{Z}}\big{\\{}(e^{-v+iu},\alpha\,e^{2k\pi
i\lambda}\,e^{-\lambda v+i\lambda
u})~{}|~{}u\in(0,2\pi),v>\textstyle\frac{\log^{+}|\alpha|}{\lambda}\big{\\}}.$

Normalizing $H_{\alpha\,e^{2k\pi i\lambda}}$ on $\tilde{L}_{\alpha\,e^{2k\pi
i\lambda}}$ avoids ambiguity. Thus

$|\\!|T|\\!|_{U}=\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha)$

for some positive measure $\mu$ on $\mathbb{C}^{*}$. Here, let us write
$|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}$ for the jacobian calculated before,

$\displaystyle|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}=\left\\{\begin{aligned}
&2(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v}),&\quad{\scriptstyle{(|\alpha|<1)}}\\\
&2(|\alpha|^{-\frac{2}{\lambda}}e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v}).&\quad{\scriptstyle{(|\alpha|\geqslant
1)}}\end{aligned}\right.$ (5)

Since $H$ is harmonic in a neighborhood of $\mathbb{H}$, it is continuous in
$\mathbb{H}$. So

$\displaystyle|\\!|T|\\!|_{U}$ $\displaystyle=\lim\limits_{\epsilon\rightarrow
0+}\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=0}^{2\pi+\epsilon}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha)$
$\displaystyle=\lim\limits_{\epsilon\rightarrow
0+}|\\!|T|\\!|_{\bigcup\limits_{k\in\mathbb{Z}}\tilde{L}_{\alpha\,e^{2k\pi
i\lambda}}}$ $\displaystyle=|\\!|T|\\!|_{\mathbb{D}^{2}}.$

Thus we can express the mass by a formula independent of the choice of
normalization

$|\\!|T|\\!|_{\mathbb{D}^{2}}=\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha).$

######  Lemma 5.2.

For each $k_{0}\in\mathbb{Z}$ fixed,

$|\\!|T|\\!|_{\mathbb{D}^{2}}=\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=2k_{0}\pi}^{2k_{0}\pi+2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha).$

###### Proof.

The disjoint union $L_{\alpha}\cap
U=\bigcup\limits_{k\in\mathbb{Z}}\tilde{L}_{\alpha\,e^{2k\pi i\lambda}}$ can
be parametrized in many other ways. For instance

$L_{\alpha}\cap
U=\bigcup\limits_{k\in\mathbb{Z}}\big{\\{}(e^{-v+iu},\alpha\,e^{2k\pi
i\lambda}\,e^{-\lambda v+i\lambda
u})~{}|~{}u\in(2k_{0}\pi,2k_{0}\pi+2\pi),v>\textstyle\frac{\log^{+}|\alpha|}{\lambda}\big{\\}}.$

By the same argument as above one concludes. ∎

###  5.1 Periodic currents, still a Fourrier series

Periodic currents behave similarly as currents in the rational case
$\lambda\in\mathbb{Q}$. Suppose $H_{\alpha}$ is periodic, i.e. there is some
$b\in\mathbb{Z}_{\geqslant 1}$ such that $H_{\alpha}(u+iv)=H_{\alpha}(u+2\pi
b+iv)$ for any $u+iv\in\mathbb{H}$. Periodic harmonic functions are
characterized as (2) of Lemma 4.1.

According to Lemma 5.2, the mass is

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=2k_{0}\pi}^{2k_{0}\pi+2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}\,du\wedge
dv\,d\mu(\alpha),$

for any $k_{0}\in\mathbb{Z}$, in particular, for $k_{0}=0,1,\dots,b-1$. Thus,
we may calculate

$\displaystyle b|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}\,du\wedge
dv\,d\mu(\alpha),$ $\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\frac{1}{b}\int_{\alpha\in\mathbb{C}^{*}}\int_{v>0}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}\,du\wedge
dv\,d\mu(\alpha),$
$\displaystyle=\frac{1}{b}\Big{\\{}\int_{|\alpha|<1}\int_{v>0}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)\,2(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\wedge
dv\,d\mu(\alpha)$ $\displaystyle\ \ \ \ \ \ \ +\int_{|\alpha|\geqslant
1}\int_{v>0}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)\,2(|\alpha|^{-\frac{2}{\lambda}}\,e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v})\,du\wedge
dv\,d\mu(\alpha)\Big{\\}},$ $\displaystyle=\frac{2\pi
b}{b}\Big{\\{}\int_{|\alpha|<1}\int_{v>0}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ +\int_{|\alpha|\geqslant
1}\int_{v>0}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2(|\alpha|^{-\frac{2}{\lambda}}\,e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)\Big{\\}},$
$\displaystyle=2\pi\,\Big{\\{}\int_{|\alpha|<1}a_{0}(\alpha)\,(1+|\alpha|^{2}\,\lambda)d\mu(\alpha)+\int_{|\alpha|\geqslant
1}a_{0}(\alpha)\,(|\alpha|^{-\frac{2}{\lambda}}+\lambda)d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \
+\int_{|\alpha|<1}b_{0}(\alpha)\,\big{(}t\frac{1}{2}+\tfrac{1}{2}|\alpha|^{2}\big{)}d\mu(\alpha)+\int_{|\alpha|<1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\big{)}d\mu(\alpha)\Big{\\}}$
$\displaystyle\approx\int_{\alpha\in\mathbb{C}^{*}}a_{0}(\alpha)\,d\mu(\alpha)+\int_{\alpha\in\mathbb{C}^{*}}b_{0}(\alpha)\,d\mu(\alpha),$

which is the same expression as in the case $\lambda\in\mathbb{Q}_{>0}$.

Next, the Lelong number is calculated as

$\displaystyle\mathscr{L}(T,0)$ $\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}$
$\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}2\pi\,\Big{\\{}\int_{|\alpha|<r^{1-\lambda}}\int_{v>-\log
r}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\,\int_{r^{1-\lambda}\leqslant|\alpha|<1}\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)\Big{\\}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\,\int_{|\alpha|\geqslant
1}\int_{v>\frac{-\log
r}{\lambda}}\big{(}a_{0}(\alpha)+b_{0}(\alpha)v\big{)}\,2\,(|\alpha|^{-\frac{2}{\lambda}}e^{-2v}+\lambda^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha)\Big{\\}}$
$\displaystyle=\lim\limits_{r\rightarrow
0+}2\pi\,\Big{\\{}\int_{|\alpha|<r^{1-\lambda}}a_{0}(\alpha)\,(1+\lambda\,|\alpha|^{2}\,r^{2\lambda-2})\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ +\int_{|\alpha|\geqslant
r^{1-\lambda}}a_{0}(\alpha)\,(|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}+\lambda)\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{|\alpha|<r^{1-\lambda}}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{2}\,r^{2\lambda-2}-\log
r-\lambda\,|\alpha|^{2}\,r^{2\lambda-2}\,\log r\big{)}\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{r^{1-\lambda}\leqslant|\alpha|<1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}-\log
r-\lambda^{-1}\,|\alpha|^{-\frac{2}{\lambda}}\,r^{2\lambda-2}\log r$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \
+\log|\alpha|+\lambda^{-1}\,|\alpha|^{-\frac{2}{\lambda}}\,\log|\alpha|\,r^{2\lambda-2}\big{)}\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ +\int_{|\alpha|\geqslant
1}b_{0}(\alpha)\,\big{(}\tfrac{1}{2}+\tfrac{1}{2}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}-2}-\log
r-\lambda^{-1}\,|\alpha|^{-\frac{2}{\lambda}}\,r^{2\lambda-2}\log
r\big{)}\,d\mu(\alpha)\Big{\\}}.$

exactly the same expression as in the case $\lambda=1$. Using the same
argument as in Lemma 4.2 one may assume that $b_{0}(\alpha)=0$ for
$\mu$-almost all $\alpha\in\mathbb{C}^{*}$. One concludes that

$\mathscr{L}(T,0)\approx\int_{\alpha\in\mathbb{C}^{*}}a_{0}(\alpha)\,d\mu(\alpha)\approx|\\!|T|\\!|_{\mathbb{D}^{2}}.$

The Lelong number is strictly positive, the same as in the case
$\lambda\in\mathbb{Q}\cup(0,1)$.

###  5.2 Non-periodic current

For periodic currents, one takes an average among $b$ expressions (4.1) in the
previous section. For non-periodic currents, there is no canonical way of
normalization. The key technique is to calculate expressions (4.1) for all
$k_{0}\in\mathbb{Z}$.

The Lelong number is expressed by

$\displaystyle\mathscr{L}(T,0)$ $\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}\Big{\\{}\int_{|\alpha|<r^{1-\lambda}}\int_{v>-\log
r}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \
+\int_{r^{1-\lambda}\leqslant|\alpha|<1}\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ +\int_{|\alpha|\geqslant
1|}\int_{v>\frac{-\log
r}{\lambda}}\int_{u=0}^{2\pi}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}du\,dv\,d\mu(\alpha)\Big{\\}}$

Recall the Poisson integral formula after multiplying a nonzero constant

$H_{\alpha}(u+iv)=\frac{1}{\pi}\,\int_{y\in\mathbb{R}}H_{\alpha}(y)\,\frac{v}{v^{2}+(y-u)^{2}}\,dy+{C}_{\alpha}\,v.$

Using the same argument as in Lemma 4.2, one may assume ${C}_{\alpha}=0$ for
all $\alpha\in\mathbb{C}^{*}$.

######  Lemma 5.3.

For any $v\geqslant\frac{1}{\lambda}>1$ and for any $u\in\mathbb{R}$,

$\displaystyle\frac{\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2}\frac{v}{v^{2}+(u-y)^{2}}e^{-2v}\Big{)}}{\frac{v}{v^{2}+(u-y)^{2}}e^{-2v}}\in\Big{(}\frac{1}{2},2\Big{)},$
$\displaystyle\frac{\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2\lambda}\frac{v}{v^{2}+(u-y)^{2}}e^{-2\lambda
v}\Big{)}}{\frac{v}{v^{2}+(u-y)^{2}}e^{-2\lambda
v}}\in\Big{(}\frac{1}{2},2\Big{)}.$

###### Proof.

This can be calculated directly,

$\displaystyle\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2}\frac{v}{v^{2}+(u-y)^{2}}e^{-2v}\Big{)}$
$\displaystyle=\Big{(}\frac{v}{v^{2}+(u-y)^{2}}+\big{(}-\frac{1}{2}\big{)}\frac{1}{v^{2}+(u-y)^{2}}+\big{(}-\frac{1}{2}\big{)}\frac{v(-2v)}{\big{(}v^{2}+(u-y)^{2}\big{)}^{2}}\Big{)}\,e^{-2v}$
$\displaystyle\frac{\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2}\frac{v}{v^{2}+(u-y)^{2}}e^{-2v}\Big{)}}{\frac{v}{v^{2}+(u-y)^{2}}e^{-2v}}$
$\displaystyle=1+\big{(}-\frac{1}{2}\frac{1}{v}\big{)}+\frac{v}{v^{2}+(u-y)^{2}}$
$\displaystyle\in\Big{(}1-\frac{1}{2v},1+\frac{1}{v}\Big{)}\subseteq\Big{(}\frac{1}{2},2\Big{)},\quad{\scriptstyle{(v>1)}},$
$\displaystyle\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2\lambda}\frac{v}{v^{2}+(u-y)^{2}}e^{-2\lambda v}\Big{)}$
$\displaystyle=\Big{(}\frac{v}{v^{2}+(u-y)^{2}}+\big{(}-\frac{1}{2\lambda}\big{)}\frac{1}{v^{2}+(u-y)^{2}}+\big{(}-\frac{1}{2\lambda}\big{)}\frac{v(-2v)}{\big{(}v^{2}+(u-y)^{2}\big{)}^{2}}\Big{)}\,e^{-2\lambda
v}$ $\displaystyle\frac{\frac{\partial}{\partial
v}\Big{(}-\frac{1}{2\lambda}\frac{v}{v^{2}+(u-y)^{2}}e^{-2\lambda
v}\Big{)}}{\frac{v}{v^{2}+(u-y)^{2}}e^{-2\lambda v}}$
$\displaystyle=1+\big{(}-\frac{1}{2\lambda}\frac{1}{v}\big{)}+\frac{1}{\lambda}\frac{v}{v^{2}+(u-y)^{2}}$
$\displaystyle\in\Big{(}1-\frac{1}{2\lambda v},1+\frac{1}{\lambda
v}\Big{)}\subseteq\Big{(}\frac{1}{2},2\Big{)},\quad{\scriptstyle{(v\geqslant\frac{1}{\lambda})}}.\qed$

######  Corollary 5.4.

For any $r$ such that $0<r\leqslant e^{-\frac{1}{\lambda}}$,

$\displaystyle\frac{1}{r^{2}}\int_{v>-\log
r}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle\approx H_{\alpha}\big{(}u+(-\log
r)i\big{)},\quad{\scriptstyle{(0<|\alpha|<r^{1-\lambda})}}$
$\displaystyle\frac{1}{r^{2}}\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle\approx H_{\alpha}\big{(}u+(\textstyle\frac{\log|\alpha|-\log
r}{\lambda})i\big{)},\quad{\scriptstyle{(0<|\alpha|<r^{1-\lambda})}}$
$\displaystyle\frac{1}{r^{2}}\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle\approx H_{\alpha}\big{(}u+(\textstyle\frac{-\log
r}{\lambda})i\big{)},\quad{\scriptstyle{(0<|\alpha|<r^{1-\lambda})}}$

Figure 13: $\frac{1}{r^{2}}$ (The integration over $v>-\log r$) $\approx$ (The
value at the boundary line $v=\log r$)

###### Proof.

When $0<r\leqslant e^{-\frac{1}{\lambda}}$, which implies $-\log
r\geqslant\frac{1}{\lambda}$, by Lemma 5.3

$\displaystyle\int_{v>-\log
r}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle=\frac{1}{\pi}\int_{v>-\log
r}\int_{y\in\mathbb{R}}H_{\alpha}(y)\frac{v}{v^{2}+(u-y)^{2}}2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda
v})dy\,dv$
$\displaystyle\approx\frac{1}{\pi}\int_{y\in\mathbb{R}}H_{\alpha}(y)\Big{\\{}\int_{v>-\log
r}\frac{\partial}{\partial
v}\Big{(}\frac{v}{v^{2}+(u-y)^{2}}\big{(}-e^{-2v}-\lambda\,|\alpha|^{2}\,e^{-2\lambda
v}\big{)}\Big{)}dv\Big{\\}}\,dy$
$\displaystyle=\frac{1}{\pi}\int_{y\in\mathbb{R}}H_{\alpha}(y)\frac{-\log
r}{(-\log
r)^{2}+(u-y)^{2}}\big{(}r^{2}+\lambda\,|\alpha|^{2}\,r^{2\lambda}\big{)}dy$
$\displaystyle=H_{\alpha}\big{(}u+(-\log
r)i\big{)}\big{(}r^{2}+\lambda\,|\alpha|^{2}\,r^{2\lambda}\big{)}$
$\displaystyle\approx r^{2}\,H_{\alpha}\big{(}u+(-\log r)i\big{)}.$

For the same reason when $r^{1-\lambda}\leqslant|\alpha|<1$ which implies
$\frac{\log|\alpha|-\log r}{\lambda}\geqslant-\log
r\geqslant\frac{1}{\lambda}$

$\displaystyle\int_{v>\frac{\log|\alpha|-\log
r}{\lambda}}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle\approx H_{\alpha}\big{(}u+(\textstyle\frac{\log|\alpha|-\log
r}{\lambda})i\big{)}\,\big{(}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}+\lambda\,r^{2})$
$\displaystyle\approx
r^{2}\,H_{\alpha}\big{(}u+(\textstyle\frac{\log|\alpha|-\log
r}{\lambda})i\big{)}.$

Last, when $|\alpha|\geqslant 1$ one has $\frac{-\log
r}{\lambda}\geqslant-\log r\geqslant\frac{1}{\lambda}$ and

$\displaystyle\int_{v>\frac{-\log
r}{\lambda}}H_{\alpha}(u+iv)|\\!|\psi_{\alpha}^{\prime}|\\!|^{2}dv$
$\displaystyle\approx H_{\alpha}\big{(}u+(\textstyle\frac{-\log
r}{\lambda})i\big{)}\,\big{(}|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}+\lambda\,r^{2})$
$\displaystyle\approx r^{2}\,H_{\alpha}\big{(}u+(\textstyle\frac{-\log
r}{\lambda})i\big{)}.\qed$

Thus

$\displaystyle\mathscr{L}(T,0)$ $\displaystyle\approx\lim\limits_{r\rightarrow
0+}\Big{\\{}\int_{|\alpha|<r^{1-\lambda}}\int_{u=0}^{2\pi}H_{\alpha}\big{(}u+(-\log
r)i\big{)}\,du\,d\mu(\alpha)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+\int_{r^{1-\lambda}\leqslant|\alpha|<1}\int_{u=0}^{2\pi}H_{\alpha}\big{(}u+(\textstyle\frac{\log|\alpha|-\log
r}{\lambda})i\big{)}\,du\,d\mu(\alpha)$ $\displaystyle\ \ \ \ \ \ \ \ \ \
+\int_{|\alpha|\geqslant
1}\int_{u=0}^{2\pi}H_{\alpha}\big{(}u+(\textstyle\frac{-\log
r}{\lambda})i\big{)}\,du\,d\mu(\alpha)\Big{\\}},$

by the inequalities (3) and (4) in the previous subsection. All terms are
positive, so the order of taking limit and integration can change

$\displaystyle\mathscr{L}(T,0)$
$\displaystyle\approx\lim\limits_{v\rightarrow+\infty}\int_{\alpha\in\mathbb{C}^{*}}\int_{u=0}^{2\pi}H_{\alpha}\big{(}u+iv\big{)}\,du\,d\mu(\alpha)$
$\displaystyle=\lim\limits_{k\rightarrow+\infty}\int_{\alpha\in\mathbb{C}^{*}}\int_{u=0}^{2\pi}\int_{y\in\mathbb{R}}H_{\alpha}(y)\frac{2k\pi}{(2k\pi)^{2}+(u-y)^{2}}\,dy\,du\,d\mu(\alpha),$

Fix some $k\in\mathbb{Z}$, $k\geqslant 2$. Define intervals $I_{N}$ for all
$N\in\mathbb{Z}$ as follows

$\displaystyle I_{0}$ $\displaystyle=[-2k\pi+2\pi,2k\pi),$ $\displaystyle
I_{N}$ $\displaystyle=\left\\{\begin{aligned} &\big{[}2k\pi
N,2k\pi(N+1)\big{)},&\quad{\scriptstyle{(N>0)}}\\\
&\big{[}2k\pi(N-1)+2\pi,2k\pi
N+2\pi\big{)},&\quad{\scriptstyle{(N<0)}}\end{aligned}\right.$

Thus $\mathbb{R}=\bigcup\limits_{N\in\mathbb{Z}}I_{N}$ is a disjoint union.

######  Lemma 5.5.

For any $u\in(0,2\pi)$ one has

$\frac{2k\pi}{(2k\pi)^{2}+(u-y)^{2}}\geqslant\frac{1}{1+(N+1)^{2}}\frac{1}{2k\pi},\quad{\scriptstyle{(y\in
I_{N})}}$

###### Proof.

Elementary. ∎

Thus

$\displaystyle\mathscr{L}(T,0)$
$\displaystyle\approx\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^{*}}\int_{u=0}^{2\pi}\int_{y\in
I_{N}}H_{\alpha}(y)\frac{2k\pi}{(2k\pi)^{2}+(u-y)^{2}}\,dy\,du\,d\mu(\alpha)$
$\displaystyle\geqslant\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^{*}}\int_{y\in
I_{N}}\int_{u=0}^{2\pi}H_{\alpha}(y)\frac{1}{1+(N+1)^{2}}\frac{1}{2k\pi}\,du\,dy\,d\mu(\alpha)$
$\displaystyle=\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\int_{\alpha\in\mathbb{C}^{*}}\int_{y\in
I_{N}}H_{\alpha}(y)\frac{1}{1+(N+1)^{2}}\frac{1}{k}\,dy\,d\mu(\alpha).$

By Lemma 5.2

$\displaystyle\int_{\alpha\in\mathbb{C}^{*}}\int_{y\in
I_{0}}H_{\alpha}(y)\,dy\,d\mu(\alpha)$
$\displaystyle=(2k-1)\,|\\!|T|\\!|_{\mathbb{D}^{2}}$ $\displaystyle\geqslant
k\,|\\!|T|\\!|_{\mathbb{D}^{2}},$
$\displaystyle\int_{\alpha\in\mathbb{C}^{*}}\int_{y\in
I_{N}}H_{\alpha}(y)\,dy\,d\mu(\alpha)$
$\displaystyle=k\,|\\!|T|\\!|_{\mathbb{D}^{2}},\quad{\scriptstyle{(N\neq
0)}}.$

Thus

$\mathscr{L}(T,0)\gtrsim\lim\limits_{k\rightarrow+\infty}\sum\limits_{N\in\mathbb{Z}}\frac{1}{1+(N+1)^{2}}\,|\\!|T|\\!|_{\mathbb{D}^{2}}\approx|\\!|T|\\!|_{\mathbb{D}^{2}}$

is nonzero.

##  6 Case $\lambda<0$. Periodic current, including all currents when
$\lambda\in\mathbb{Q}_{<0}$

For any $\alpha\in\mathbb{C}^{*}$ fixed, the leaf $L_{\alpha}$ is contained in
a real 3-dimensional analytic Levi-flat CR manifold
$|w|=|\alpha|\,|z|^{\lambda}$, which can be viewed as a curve in $|z|,|w|$
coordinates. The norms $|z|$ and $|w|$ depends only on $v$. No leaf
$L_{\alpha}$ tends to the singularity $(0,0)$. For $r$ sufficiently small, the
leaf $L_{\alpha}$ is outside of $r\,\mathbb{D}^{2}$.

Figure 14: Case $\lambda<0$

Like the case $\lambda>0$, when one fixes $|z|=r$ for some $r\in(0,1)$, then
$|w|=|\alpha|\,|z|^{\lambda}$ is uniquely determined and the real
2-dimensional leaf $L_{\alpha}$ becomes a real 1-dimensional curve
$L_{\alpha,r}:=L_{\alpha}\cap\mathbb{T}^{2}_{r}$ on the torus
$\mathbb{T}^{2}_{r}:=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}|z|=r,|w|=|\alpha|\,r^{\lambda}\\}$.
It is a closed curve if $\lambda\in\mathbb{Q}$, a dense curve on
$\mathbb{T}^{2}_{r}$ if $\lambda\notin\mathbb{Q}$.

Let $T$ be a harmonic current directed by $\mathscr{F}$. Then
$T|_{P_{\alpha}}$ has the form $h_{\alpha}(z,w)[P_{\alpha}]$. Let
$H_{\alpha}:=h_{\alpha}\circ\psi_{\alpha}(u+iv)$. It is a positive harmonic
function for $\mu$-almost all $\alpha\in\mathbb{D}^{*}$ defined on a
neighborhood of a horizontal strip
$\\{(u,v)\in\mathbb{R}^{2}~{}|~{}0<v<\frac{\log|\alpha|}{\lambda}\\}$.

Like what happened in the case $\lambda>0$, one only calculates the mass on an
open subset $U:=\\{(z,w)\in\mathbb{D}^{2}~{}|~{}z\notin\mathbb{R}_{\geqslant
0},w\neq 0\\}$. For each $\alpha\in\mathbb{D}^{*}$ one normalizes $H_{\alpha}$
by setting $H_{\alpha}(0)=1$ to fix the expression of $T:=\int
h_{\alpha}[P_{\alpha}]d\mu(\alpha)$. Similar to Lemma 5.2, for each
$k_{0}\in\mathbb{Z}$ fixed,

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\int_{0<|\alpha|<1}\int_{v=0}^{\frac{\log|\alpha|}{\lambda}}\int_{u=2\,k_{0}\,\pi}^{2\,k_{0}\,\pi+2\,\pi}H_{\alpha}(u+iv)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\,dv\,d\mu(\alpha)$
$\displaystyle\mathscr{L}(T,0)$ $\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}$
$\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}\int_{0<|\alpha|<r^{1-\lambda}}\int_{v=-\log
r}^{\frac{\log|\alpha|-\log
r}{\lambda}}\int_{u=2\,k_{0}\,\pi}^{2\,k_{0}\,\pi+2\,\pi}H_{\alpha}(u+iv)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\,dv\,d\mu(\alpha)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
dy\,dx\,d\mu(\alpha)$

Now assume $T$ is periodic, we treat Theorem 1.5. Suppose that there exists
some $b\in\mathbb{Z}_{\leqslant 1}$ such that
$H_{\alpha}(u+iv)=H_{\alpha}(u+2\pi b+iv)$ for all $\alpha\in\mathbb{D}^{*}$
and all $(u,v)$ in a neighborhood of the strip
$\\{(u+iv)\in\mathbb{C}~{}|~{}u\in\mathbb{R},v\in[0,\frac{\log|\alpha|}{\lambda}]\\}$.
Like Lemma 4.1, one proves

######  Lemma 6.1.

Let $F(u,v)$ be a positive harmonic function on a neighborhood of the
horizontal strip $\\{(u+iv)\in\mathbb{C}~{}|~{}u\in\mathbb{R},v\in[0,C]\\}$
for some $C>0$. Suppose $F(u,v)=F(u+2\pi b,v)$ on this strip. Then

$F(u,v)=\sum\limits_{k\in\mathbb{Z},k\neq
0}\big{(}a_{k}\,e^{\frac{kv}{b}}\cos(\tfrac{ku}{b})+b_{k}\,e^{\frac{kv}{b}}\sin(\tfrac{ku}{b})\big{)}+a_{0}\,(1-C^{-1}\,v)+b_{0}\,v,$

for some $a_{k},b_{k}\in\mathbb{R}$ with $a_{0}\geqslant 0$ and
$b_{0}\geqslant 0$.

###### Proof.

The proof is almost the same as that of Lemma 4.1. Using Fourier series and
calculating Laplacian, one concludes that

$F(u,v)=\sum\limits_{k\in\mathbb{Z},k\neq
0}\big{(}a_{k}\,e^{\frac{kv}{b}}\cos(\tfrac{ku}{b})+b_{k}\,e^{\frac{kv}{b}}\sin(\tfrac{ku}{b})\big{)}+p+q\,v,$

for some $a_{k},b_{k},p,q\in\mathbb{R}$. For any $v\in[0,C]$, $F(u,v)\geqslant
0$ implies

$\int_{u=0}^{2\pi b}F(u,v)du=2\pi b\,(p+q\,v)\geqslant 0.$

Thus $p\geqslant 0$ and $q\geqslant-C^{-1}\,p$. One may write
$p+q\,v=p\,(1-C^{-1}\,v)+(q+C^{-1}\,p)\,v$ with $p=:a_{0}\geqslant 0$ and
$q+C^{-1}\,p=:b_{0}\geqslant 0$.∎

For periodic currents one may assume

$\displaystyle H_{\alpha}(u+iv)=\sum\limits_{k\in\mathbb{Z},k\neq
0}\big{(}a_{k}(\alpha)\,e^{\frac{kv}{b}}\cos(\tfrac{ku}{b})+b_{k}(\alpha)\,e^{\tfrac{kv}{b}}\sin(\tfrac{ku}{b})\big{)}+a_{0}(\alpha)\,(1-\textstyle\frac{\lambda}{\log|\alpha|}\,v)+b_{0}(\alpha)\,v,$
(6)

for some $a_{k}(\alpha),b_{k}(\alpha)\in\mathbb{R}$ with
$a_{0}(\alpha)\geqslant 0$ and $b_{0}(\alpha)\geqslant 0$. According to Lemma
5.2, for any $k_{0}\in\mathbb{Z}$, use the jacobian (5)

$|\\!|T|\\!|_{\mathbb{D}^{2}}=\int_{0<|\alpha|<1}\int_{v=0}^{\frac{\log|\alpha|}{\lambda}}\int_{u=2k_{0}\pi}^{2k_{0}\pi+2\pi}H_{\alpha}(u+iv)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\,dv\,d\mu(\alpha).$

Next, using $0=\int_{0}^{2\pi b}\cos(\frac{ku}{b})du$ for $k\neq 0$ and the
same for $\sin(\frac{ku}{b})$, let us calculate the average among
$k_{0}=0,1,\dots,b-1$ for the mass

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=\frac{1}{b}\int_{0<|\alpha|<1}\int_{v=0}^{\frac{\log|\alpha|}{\lambda}}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\,dv\,d\mu(\alpha)$
$\displaystyle=\frac{2\pi
b}{b}\int_{0<|\alpha|<1}\int_{v=0}^{\frac{\log|\alpha|}{\lambda}}\left(a_{0}(\alpha)\,(1-\textstyle\frac{\lambda}{\log|\alpha|}\,v)+b_{0}(\alpha)\,v\right)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha),$

and for the Lelong number

$\displaystyle\mathscr{L}(T,0)$ $\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{r^{2}}|\\!|T|\\!|_{r\mathbb{D}^{2}}$
$\displaystyle=\lim\limits_{r\rightarrow
0+}\frac{1}{b\,r^{2}}\int_{0<|\alpha|<r^{1-\lambda}}\int_{v=-\log
r}^{\frac{\log|\alpha|-\log r}{\lambda}}\int_{u=0}^{2\pi
b}H_{\alpha}(u+iv)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,du\,dv\,d\mu(\alpha)$
$\displaystyle=\lim\limits_{r\rightarrow 0+}\frac{2\pi
b}{b\,r^{2}}\int_{0<|\alpha|<r^{1-\lambda}}\int_{v=-\log
r}^{\frac{\log|\alpha|-\log
r}{\lambda}}\left(a_{0}(\alpha)\,(1-\textstyle\frac{\lambda}{\log|\alpha|}\,v)+b_{0}(\alpha)\,v\right)\,2\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv\,d\mu(\alpha).$

Introduce the two functions of $r\in(0,1]$ given by elementary integrals

$\displaystyle I_{a}(r)$ $\displaystyle:=\frac{1}{r^{2}}\int_{v=-\log
r}^{\frac{\log|\alpha|-\log
r}{\lambda}}2\,(1-\textstyle\frac{\lambda}{\log|\alpha|}\,v)\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv,$
$\displaystyle=1+\lambda\,|\alpha|^{2}\,r^{2\lambda-2}+\frac{1}{2\log|\alpha|}\left(-2|\alpha|^{-\frac{2}{\lambda}}r^{\frac{2}{\lambda}-2}\log(r)+\lambda|\alpha|^{-\frac{2}{\lambda}}r^{\frac{2}{\lambda}-2}+2\,\lambda^{2}\,|\alpha|^{2}\,r^{2\lambda-2}\log(r)-\lambda\,|\alpha|^{2}\,r^{2\lambda-2}\right),$
$\displaystyle I_{b}(r)$ $\displaystyle:=\frac{1}{r^{2}}\int_{v=-\log
r}^{\frac{\log|\alpha|-\log
r}{\lambda}}2\,v\,(e^{-2v}+\lambda^{2}\,|\alpha|^{2}\,e^{-2\lambda\,v})\,dv$
$\displaystyle=\frac{1}{2}\left(-\frac{|\alpha|^{-\frac{2}{\lambda}}r^{\frac{2}{\lambda}-2}(\lambda+2\log|\alpha|-2\log(r))}{\lambda}+|\alpha|^{2}r^{2\lambda-2}(1-2\lambda\log(r))-2\log|\alpha|\right),$

to describe the contributions from $a_{0}(\alpha)$ part and from
$b_{0}(\alpha)$ part.

Figure 15: The green = the blue dotted (gives $I_{a}(r)$) + the yellow dashed
(gives $I_{b}(r)$)

Then we can express

$\displaystyle|\\!|T|\\!|_{\mathbb{D}^{2}}$
$\displaystyle=2\,\pi\,\int_{0<|\alpha|<1}\Big{(}a_{0}(\alpha)\,I_{a}(1)+b_{0}(\alpha)\,I_{b}(1)\Big{)}\,d\mu(\alpha),$
$\displaystyle\mathscr{L}(T,0)$
$\displaystyle=2\,\pi\,\lim\limits_{r\rightarrow
0+}\int_{0<|\alpha|<r^{1-\lambda}}\Big{(}a_{0}(\alpha)\,I_{a}(r)+b_{0}(\alpha)\,I_{b}(r)\Big{)}\,d\mu(\alpha).$

Observe that

$\displaystyle I_{a}(1)$
$\displaystyle=1+\lambda\,|\alpha|^{2}+\frac{\lambda\left(|\alpha|^{-\frac{2}{\lambda}}-|\alpha|^{2}\right)}{2\log|\alpha|},$
$\displaystyle I_{b}(1)$
$\displaystyle=\frac{1}{2}\left(-\frac{|\alpha|^{-\frac{2}{\lambda}}(\lambda+2\log|\alpha|)}{\lambda}+|\alpha|^{2}-2\log|\alpha|\right).$

Fix any $\alpha\in\mathbb{D}^{*}$, by definition $r^{2}I_{a}(r)$ and
$r^{2}I_{b}(r)$ are increasing for $r\in(0,1]$, since the interval of
integration $(-\log r,\frac{\log|\alpha|-\log r}{\lambda})$ is expanding and
the function integrated is positive. In particular, for any $r\in(0,1]$,

$I_{a}(r)\leqslant r^{-2}\,I_{a}(1),\ \ \ \ I_{b}(r)\leqslant
r^{-2}\,I_{b}(1).$

It is more subtle to talk about monotonicity of $I_{a}(r)$ and $I_{b}(r)$. We
expect upper bounds of $I_{a}(r)/I_{a}(1)$ and $I_{b}(r)/I_{b}(1)$ for
$r\in(0,1]$ which are independent of $\alpha$, i.e. depend only on $\lambda$.

######  Lemma 6.2.

For any $r\in(0,1)$ and any $\alpha\in\mathbb{C}$ with
$0<|\alpha|<r^{1-\lambda}<1$, one has

$0<I_{a}(r)<I_{a}(1).$

###### Proof.

A differentiation gives

$\displaystyle\frac{d}{dr}I_{a}(r)$
$\displaystyle=\underbrace{\frac{|\alpha|^{-\frac{2}{\lambda}}}{\lambda\,r^{3}\,\log|\alpha|}}_{>0}\Big{(}\lambda^{2}\big{(}|\alpha|^{2+\frac{2}{\lambda}}r^{2\lambda}-r^{\frac{2}{\lambda}}\big{)}-2(1-\lambda)\big{(}\lambda^{3}|\alpha|^{2+\frac{2}{\lambda}}r^{2\lambda}+r^{\frac{2}{\lambda}}\big{)}\log(r)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-2(1-\lambda)\lambda^{2}|\alpha|^{2+\frac{2}{\lambda}}r^{2\lambda}\log|\alpha|\Big{)}.$

It suffices to show that $\frac{d}{dr}I_{a}(r)>0$ when $r\in(0,1)$ and
$0<|\alpha|<r^{1-\lambda}$.

Introduce the new variable $t:=\frac{|\alpha|}{r^{1-\lambda}}\in(0,1)$. In the
big parentheses, replace $|\alpha|$ by $t\,r^{1-\lambda}$ and $\log|\alpha|$
by $\log(t)+(1-\lambda)\log(r)$

$\displaystyle\frac{d}{dr}I_{a}(r)$
$\displaystyle=\underbrace{\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda
r^{3}\log|\alpha|}}_{>0}\Big{(}\lambda^{2}(t^{2+\frac{2}{\lambda}}-1)-2\,(1-\lambda)\,(t^{2+\frac{2}{\lambda}}+1)\log(r)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\underbrace{-2\,(1-\lambda)\,\lambda^{2}\,t^{2+\frac{2}{\lambda}}\log(t)}_{>0}\Big{)}$
$\displaystyle>\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda
r^{3}\log|\alpha|}\Big{(}\lambda^{2}\,\underbrace{(t^{2+\frac{2}{\lambda}}-1)}_{\geqslant
0}\underbrace{-2\,(1-\lambda)\,(t^{2+\frac{2}{\lambda}}+1)\log(r)}_{>0}\Big{)}>0,$

since $\lambda\in[-1,0)$ implies $t^{2+\frac{2}{\lambda}}\geqslant 1$.∎

It is not true that $I_{b}(r)$ is increasing on $(0,1]$, but on a smaller
half-neighborhood of $0$, independent of $\alpha$, it is increasing. This
suffice to give an upper bound of $I_{b}(r)/I_{b}(1)$.

######  Lemma 6.3.

For any $r\in(0,e^{\frac{1}{2\,\lambda\,(1-\lambda)}})$ and any
$\alpha\in\mathbb{C}$ with $0<|\alpha|<r^{1-\lambda}<1$, one has

$0<I_{b}(r)<I_{b}(e^{\frac{1}{2\lambda(1-\lambda)}})\leqslant
e^{\frac{1}{-\lambda\,(1-\lambda)}}\,I_{b}(1).$

###### Proof.

A differentiation gives

$\displaystyle\frac{d}{dr}I_{b}(r)$
$\displaystyle=\underbrace{\frac{|\alpha|^{-\frac{2}{\lambda}}}{\lambda^{2}r^{3}}}_{>0}\Big{(}-\lambda^{2}\big{(}|\alpha|^{2+\frac{2}{\lambda}}r^{2\lambda}-r^{\frac{2}{\lambda}}\big{)}+2(1-\lambda)\big{(}\lambda^{3}|\alpha|^{2+\frac{2}{\lambda}}r^{2\lambda}+r^{\frac{2}{\lambda}}\big{)}\log(r)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-2(1-\lambda)r^{\frac{2}{\lambda}}\log|\alpha|\Big{)}$

It suffices to show that $\frac{d}{dr}I_{b}(r)>0$ when
$0<r<e^{\frac{1}{2\,\lambda\,(1-\lambda)}}$ and $0<|\alpha|<r^{1-\lambda}$.

Again, introduce the variable $t:=\frac{|\alpha|}{r^{1-\lambda}}\in(0,1)$ and
replace $\alpha$ and $\log|\alpha|$ in the parentheses

$\displaystyle\frac{d}{dr}I_{b}(r)$
$\displaystyle=\underbrace{\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda^{2}\,r^{3}}}_{>0}\Big{(}-\lambda^{2}\,(t^{2+\frac{2}{\lambda}}-1)+2\,\lambda\,(1-\lambda)\,(\lambda^{2}\,t^{2+\frac{2}{\lambda}}+1)\,\log(r)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\underbrace{-2(1-\lambda)\log(t)}_{>0}\Big{)}$
$\displaystyle>\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda^{2}\,r^{3}}\Big{(}-\lambda^{2}\,(t^{2+\frac{2}{\lambda}}-1)+\underbrace{2\,\lambda\,(1-\lambda)\,(\lambda^{2}\,t^{2+\frac{2}{\lambda}}+1)}_{<0}\,\underbrace{\log(r)}_{<\frac{1}{2\lambda(1-\lambda)}<0}\Big{)}$
$\displaystyle>\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda^{2}\,r^{3}}\Big{(}-\lambda^{2}\,(t^{2+\frac{2}{\lambda}}-1)+\lambda^{2}\,t^{2+\frac{2}{\lambda}}+1\Big{)}=\frac{|\alpha|^{-\frac{2}{\lambda}}\,r^{\frac{2}{\lambda}}}{\lambda^{2}\,r^{3}}\big{(}\lambda^{2}\,+1\big{)}>0.\qed$

###### End of proof of Theorem 1.5.

From what precedes, the Lelong number is zero

$\displaystyle\mathscr{L}(T,0)$
$\displaystyle=2\,\pi\,\lim\limits_{r<e^{\frac{1}{2\lambda(1-\lambda)}},r\rightarrow
0+}\int_{0<|\alpha|<r^{1-\lambda}}\big{(}a_{0}(\alpha)\,I_{a}(r)+b_{0}(\alpha)\,I_{b}(r)\big{)}\,d\mu(\alpha)$
$\displaystyle<2\,\pi\,\lim\limits_{r\rightarrow
0+}\int_{0<|\alpha|<r^{1-\lambda}}\big{(}a_{0}(\alpha)\,I_{a}(1)+b_{0}(\alpha)\,e^{\frac{1}{-2\,\lambda\,(1-\lambda)}}\,I_{b}(1)\big{)}\,d\mu(\alpha)$
$\displaystyle\approx 2\,\pi\,\lim\limits_{r\rightarrow
0+}\int_{0<|\alpha|<r^{1-\lambda}}\big{(}a_{0}(\alpha)\,I_{a}(1)+b_{0}(\alpha)\,I_{b}(1)\big{)}\,d\mu(\alpha)=0,$

since
$|\\!|T|\\!|_{\mathbb{D}^{2}}=2\,\pi\,\int_{0<|\alpha|<1}\big{(}a_{0}(\alpha)\,I_{a}(1)+b_{0}(\alpha)\,I_{b}(1)\big{)}\,d\mu(\alpha)$
is finite.∎

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Zhangchi Chen, Université Paris-Saclay, CNRS, Laboratoire de mathématiques
d’Orsay, 91405, Orsay, France
<EMAIL_ADDRESS>https://www.imo.universite-paris-
saclay.fr/$\sim$chen/


# Equal Sum and Product Problem III

,  Csaba Sándor<EMAIL_ADDRESS>Department of Stochastics, Institute
of Mathemetics, Budapest University of Technology and Economics, Műegyetem
rkp. 3., H-1111 Budapest, Hungary;
Department of Computer Science and Information Theory, Budapest University of
Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary;
MTA-BME Lendület Arithmetic Combinatorics Research Group, Műegyetem rkp. 3.,
H-1111 Budapest, Hungary.  and  Maciej Zakarczemny<EMAIL_ADDRESS>Department of Applied Mathematics, Cracow University of Technology, Warszawska
24, 31-155 Kraków, Poland.

(Date: May 2024)

###### Abstract.

Denote by $N(n)$ the number of integer solutions
$(x_{1},\,x_{2},\ldots,x_{n})$ of the equation
$x_{1}+x_{2}+\ldots+x_{n}=x_{1}x_{2}\cdot\ldots\cdot x_{n}$ such that
$x_{1}\geq x_{2}\geq\ldots\geq x_{n}\geq 1$, $n\in\mathbb{Z}^{+}$. The aim of
this paper are is twofold: first we present an asymptotic formula for
$\sum\limits_{2\leq n\leq x}N(n)$, then we verify that the counting function
$N(n)$ takes very large value compared to its average value.

## Introduction

The Equal-Sum-Product Problem is a difficult and unsolved problem in number
theory; see Guy [4] and Schinzel [14]. To formulate the problem, we introduce
the following notation. Let $n$ be a positive integer, $n\geq 2$. Denote by
$N(n)$ the number of integer solutions $(x_{1},\,x_{2},\ldots,x_{n})$ of the
equation

(1.1) $x_{1}+x_{2}+\ldots+x_{n}=x_{1}x_{2}\cdot\ldots\cdot x_{n}$

such that $x_{1}\geq x_{2}\geq\ldots\geq x_{n}\geq 1$, $n\in\mathbb{Z}^{+}$.
Throughout this paper $x$ and $x_{i}$ denote positive integer, the term
”almost all” means with the exception of the set of natural density zero
(asymptotic density zero).

The Diophantine equation (1.1) has at least one solution: $(n,2,1,\dots,1)$.
Let us call an integer exceptional if there is no other solution, that is
$N(n)=1$.

######  Conjecture 1.1.

The set of exceptional values is finite:

$E=\\{2,3,4,6,24,114,174,444\\}.$

Conjecture 1.1 has been proposed by many authors (see [6, 1, 12, 4]). It is
known that there is no other solution up to $10^{10}$ (see [2]).

Schinzel [13, 14] asked whether the number $N(n)$ tends to infinity with $n$.

######  Problem 1.2.

Is it true that $\displaystyle\lim_{n\to\infty}N(n)=\infty$?

Other related problems and results in connection with equation (1.1) can be
found in the papers [2, 5, 8, 13, 14, 16, 17].

The aim of this paper are is twofold: first we present an asymptotic formula
for $\sum\limits_{2\leq n\leq x}N(n)$, then we verify that the counting
function $N(n)$ takes very large value compared to its average value.

##  2\. Main Results

Denote by

$f(x)=|\\{(y_{1},y_{2},\ldots,y_{k}):y_{1}\geq y_{2}\geq\ldots\geq y_{k}\geq
2,\,y_{i}\in\mathbb{Z},\,k\in\mathbb{Z}^{+},y_{1}y_{2}\cdot\ldots\cdot
y_{k}\leq x\\}|.$

If $x\geq 2$, $x\in\mathbb{Z}$, then $f(x)+1$ represents the number of
partitions up to $x$ with a product smaller or equal to $x$ (see [9]).

Let $a_{n}$ denote the number of distinct unordered factorisations of the
positive integer $n$ into the factors larger than $1$, where the order of
factors does not matter (see [10]). Hence

(2.1) $f(x)=\sum_{2\leq n\leq x}a_{n}.$

Oppenheim [11] determined the asymptotic behaviour of $f(x)$:

(2.2) $f(x)\sim\frac{xe^{2\sqrt{\log x}}}{2\sqrt{\pi}(\log x)^{3/4}}.$

This result was independently obtained by Szekeres and Turán ([15]).
We prove a similar result for the sequence $N(n)$.

######  Theorem 2.1.

We have the asymptotics

$\sum_{n\leq x}N(n)\sim\frac{xe^{2\sqrt{\log x}}}{2\sqrt{\pi}(\log x)^{3/4}}.$

The second author proved that for every $k\in\mathbb{Z}^{+}$ and for almost
all $n$ we have $N(n)>k$. By Theorem 2.1 we get the following stronger result.

######  Theorem 2.2.

Let $\varepsilon>0$. Then for almost all $n$ we have

$(\log n)^{\log 2-\varepsilon}<N(n)<e^{2\sqrt{\log n}}.$

Let $\varepsilon>0$. Then $\frac{xe^{2\sqrt{\log x}}}{2\sqrt{\pi}(\log
x)^{3/4}}<x^{1+\varepsilon}$ for $x>x_{0}(\varepsilon)$. Hence we have the
following consequence of Theorem 2.1.

######  Corollary 2.3.

Let $\varepsilon>0$. There exists a positive integer $N_{0}(\varepsilon)$ such
that

$N(n)<n^{1+\varepsilon}$

for $n\geq N_{0}(\varepsilon)$.

Let $\varepsilon>0$. In 1983, Canfield, Erdős and Pomerance proved [3] that

(2.3) $a_{n}\leq n^{1-(1-\varepsilon)\frac{\log\log\log n}{\log\log n}}$

if $n$ is large enough. On the other hand, for infinitely many $n$,

(2.4) $a_{n}\geq n^{1-(1+\varepsilon)\frac{\log\log\log n}{\log\log n}}.$

We prove a similar lower bound for the sequence $N(n)$.

######  Theorem 2.4.

Let $\varepsilon>0$. For infinitely many $n$,

(2.5) $N(n)\geq n^{1-(3+\varepsilon)\frac{\log\log\log n}{\log\log n}}.$

######  Corollary 2.5.

Let $\varepsilon>0$. There exist infinitely many $n$ such that

(2.6) $N(n)>n^{1-\varepsilon}.$

##  3\. Proofs

The proof of Theorem 2.1 is a consequence of the following lemma.

######  Lemma 3.1.

Let $x\geq 2$. For any function $1\leq g(x)\leq 2x$, we have

(3.1) $f(x)-x<\sum\limits_{2\leq n\leq x}N(n)\leq
f(x+g(x))+xf(\tfrac{2x(1+\log_{2}x)}{g(x)}).$

###### Proof.

If $x_{1}+\ldots+x_{n}=x_{1}\cdot\ldots\cdot x_{n},\,n\geq 2$, where
$x_{1}\geq x_{2}\geq\ldots\geq x_{k}\geq 2$, $x_{k+1}=\ldots=x_{n}=1$, then
$x_{1}\cdot\ldots\cdot x_{k}=x_{1}+\ldots+x_{k}+n-k,$ that is

$n=x_{1}\cdot\ldots\cdot x_{k}-(x_{1}+\ldots+x_{k})+k.$

The condition $n\geq 2$ implies $k\geq 2$. It follows that

(3.2) $\begin{split}N(n)=|\\{(x_{1},\ldots,x_{k}):&\,x_{1}\geq\ldots\geq
x_{k}\geq 2,k\geq 2,\\\ &x_{1}\cdot\ldots\cdot
x_{k}-(x_{1}+\ldots+x_{k})+k=n\\}|.\end{split}$

If $x_{1}\cdot\ldots\cdot x_{k}\leq x$, then $2\leq x_{1}\cdot\ldots\cdot
x_{k}-(x_{1}+\ldots+x_{k})+k\leq x$. Hence,

$\displaystyle\sum\limits_{2\leq n\leq x}N(n)$
$\displaystyle=|\\{(x_{1},\ldots,x_{k}):x_{1}\geq\ldots\geq x_{k}\geq 2,k\geq
2,$ $\displaystyle\hskip 79.6678pt2\leq x_{1}\cdot\ldots\cdot
x_{k}-(x_{1}+\ldots+x_{k})+k\leq x\\}|$
$\displaystyle\geq|\\{(x_{1},\ldots,x_{k}):x_{1}\geq x_{2}\geq\ldots\geq
x_{k}\geq 2,k\geq 2,\,x_{1}\cdot\ldots\cdot x_{k}\leq x\\}|$
$\displaystyle=|\\{(x_{1},\ldots,x_{k}):x_{1}\geq x_{2}\geq\ldots\geq
x_{k}\geq 2,\,k\geq 1,\,x_{1}\cdot\ldots\cdot x_{k}\leq x\\}|$
$\displaystyle\hskip 11.38092pt-(x-1)$ $\displaystyle>f(x)-x$

On the other hand, for any function $g(x)$:

$\displaystyle\sum\limits_{2\leq n\leq x}N(n)$
$\displaystyle=|\\{(x_{1},\ldots,x_{k}):\,x_{1}\geq\ldots\geq x_{k}\geq
2,\,k\geq 2,$ $\displaystyle\hskip 82.51282pt2\leq x_{1}\cdot\ldots\cdot
x_{k}-(x_{1}+\ldots+x_{k})+k\leq x,$ $\displaystyle\hskip
82.51282ptx_{1}+\ldots+x_{k}<g(x)\\}|$
$\displaystyle+|\\{(x_{1},\ldots,x_{k}):\,x_{1}\geq\ldots\geq x_{k}\geq
2,\,k\geq 2,$ $\displaystyle\hskip 79.6678pt2\leq x_{1}\cdot\ldots\cdot
x_{k}-(x_{1}+\ldots+x_{k})+k\leq x,$ $\displaystyle\hskip
79.6678ptx_{1}+\ldots+x_{k}\geq g(x)\\}|.$

If $x_{1}+\ldots+x_{k}<g(x)$, then $x_{1}\cdot\ldots\cdot x_{k}\leq
x+(x_{1}+\ldots+x_{k})-k<x+g(x)$.
Thus,

$\displaystyle\hskip 14.22636pt|\\{(x_{1},\ldots,x_{k}):\,x_{1}\geq\ldots\geq
x_{k}\geq 2,\,k\geq 2,$ $\displaystyle\hskip 82.51282pt2\leq
x_{1}\cdot\ldots\cdot x_{k}-(x_{1}+\ldots+x_{k})+k\leq x,$
$\displaystyle\hskip 82.51282ptx_{1}+\ldots+x_{k}<g(x)\\}|$ $\displaystyle\leq
f(x+g(x)).$

If $x_{1}+\ldots+x_{n}=x_{1}\cdot\ldots\cdot x_{n},\,x_{1}\geq\ldots\geq
x_{n}\geq 1,\,x_{i}\in\mathbb{Z},$ then $x_{1}\cdot\ldots\cdot x_{n}\leq 2n$,
(see [5, Theorem 6]). It follows that $x_{1}\cdot\ldots\cdot
x_{k}=x_{1}\cdot\ldots\cdot x_{n}\leq 2n\leq 2x$. Since $k\geq 2$, we have
$x_{1}\leq x$. Therefore we have at most $x$ possibilities for $x_{1}$. Note
also that $2^{k}\leq x_{1}\cdot\ldots\cdot x_{k}\leq 2n$ implies $k\leq
1+\log_{2}n\leq 1+\log_{2}x.$

If $x_{1}+\ldots+x_{k}\geq g(x),$ then
$x_{1}\geq\tfrac{g(x)}{k}\geq\tfrac{g(x)}{1+\log_{2}x}$. Thus,

(3.3) $x_{2}\cdot\ldots\cdot
x_{k}\leq\tfrac{2x}{x_{1}}\leq\tfrac{2x(1+\log_{2}x)}{g(x)}.$

Therefore, we have at most $f\left(\tfrac{2x(1+\log_{2}x)}{g(x)}\right)$
possibilities for $(x_{2},\ldots,x_{k})$. Hence

$\displaystyle\hskip 14.22636pt|\\{(x_{1},\ldots,x_{k}):\,x_{1}\geq\ldots
x_{k}\geq 2,\,x_{i}\in\mathbb{Z},\,k\geq 2,$ $\displaystyle\hskip
82.51282pt2\leq x_{1}\cdot\ldots\cdot x_{k}-(x_{1}+\ldots+x_{k})+k\leq x,$
$\displaystyle\hskip 82.51282ptx_{1}+\ldots+x_{k}\geq g(x)\\}|$
$\displaystyle\leq xf\left(\tfrac{2x(1+\log_{2}x)}{g(x)}\right),$

which completes the proof. ∎

###### Proof of Theorem 2.1.

Consider $g(x)=\frac{x}{e^{\sqrt{\log x}}}$ in Lemma 3.1. ∎

###### Proof of Theorem 2.2.

Let $\varepsilon>0$. Let $d(n)$ denote the number of positive divisors of the
positive integer $n$. We have $N(n)\geq\tfrac{1}{2}d(n-1)$, see [16, Corollary
2.2]. It is well known that for almost all $n$ we have $d(n)>(\log n)^{\log
2-\varepsilon}$, see [7]. It follows that

$N(n)>\tfrac{1}{2}d(n-1)\geq(\log(n-1))^{\log 2-\frac{\varepsilon}{2}}>(\log
n)^{\log 2-\varepsilon}$

for almost all positive integer $n$.

On the other hand if $f(N)=|\\{n:\,N(n)\geq e^{2\sqrt{\log n}},\,2\leq n\leq
N\\}|$, then

$(1+o(1))\frac{Ne^{2\sqrt{\log N}}}{2\sqrt{\pi}(\log N)^{3/4}}=\sum_{2\leq
n\leq N}N(n)\geq\sum_{\frac{N}{\log N}\leq n\leq N,N(n)\geq e^{2\sqrt{\log
n}}}N(n)\geq$ $(f(N)-\frac{N}{\log N})e^{2\sqrt{\log\frac{N}{\log
N}}}=(1+o(1))(f(N)-\frac{N}{\log N})e^{2\sqrt{\log N}}.$

It follows that

$f(N)\leq(\frac{1}{2\sqrt{\pi}}+o(1))\frac{N}{(\log N)^{3/4}}.$

Thus, $f(N)=o(N),$ hence for almost all $n$ we have $N(n)<e^{2\sqrt{\log n}}$.
∎

###### Proof of Theorem 2.4.

Let $m$ be a positive integer. Denote by $p_{m}$ the $m$-th prime number. Let
$P_{m}=p_{1}p_{2}\cdot\ldots\cdot p_{m}$, $P_{0}=1$. By the Prime Number
Theorem, $p_{m}=(1+o(1))m\log m$. Hence,

(3.4) $P_{m}=e^{\sum_{k=1}^{m}\log p_{k}}=e^{\sum_{k=2}^{m}(\log k+\log\log
k+o(1))}.$

We have (see also [7, Theorem 6.3]),

(3.5) $\begin{split}\sum_{k=2}^{m}\log
k=&\sum_{k=2}^{m}\left(\int_{k}^{k+1}\log
xdx+O\left(\log(k+1)-\log(k)\right)\right)\\\ =&\int_{2}^{m+1}\log xdx+O(\log
m)\\\ =&m\log m-m+O(\log m).\end{split}$

Analogously,

(3.6) $\begin{split}\sum_{k=2}^{m}\log\log
k=&\sum_{k=2}^{m}\left(\int_{k}^{k+1}\log\log
xdx+O\left(\log\log(k+1)-\log\log k\right)\right)\\\ =&\int_{2}^{m+1}\log\log
xdx+O(\log\log m)\\\ =&\int_{2}^{m+1}\log\log x+\frac{1}{\log
x}dx-\int_{2}^{m+1}\frac{1}{\log x}dx+O(\log\log m)\\\ =&m\log\log
m+O\left(\frac{m}{\log m}\right).\end{split}$

By (3.4), (3.5), (3.6) we obtain

(3.7) $P_{m}=e^{m\log m+m\log\log m-m+o(m)}\,\,\mathrm{as}\,\,m\to\infty.$

One may notice that the following equality holds:

(3.8) $\frac{P_{(i+1)m}}{P_{im}}=e^{m\log m+o(m\log m)}$

as $i=O(\log m)$ and $m\to\infty$.
Let us suppose that $k\geq 3$. By 3.7

$P_{k\lfloor\log k\rfloor}=e^{k\log^{2}k+(2+o(1))k\log k\log\log k}.$

Let

(3.9) $M_{k}=\max_{0\leq i\leq\lfloor\log
k\rfloor}\frac{P_{(i+1)k}}{P_{ik}}=e^{k\log k+o(k\log k)}.$

Denote by $B_{n}$ be the number of partitions of a set of size $n$, which are
called Bell numbers. We will show that

(3.10) $\sum_{P_{k\lfloor\log k\rfloor}-\lfloor\log k\rfloor M_{k}<n\leq
P_{k\lfloor\log k\rfloor}}N(n)\geq B_{k}^{\lfloor\log k\rfloor}.$

It is well known that $B_{n}=e^{n\log n-n\log\log n-n+o(n)}$ as $n\to\infty$.
Hence

(3.11) $\max_{P_{k\lfloor\log k\rfloor}-\lfloor\log k\rfloor M_{k}<n\leq
P_{k\lfloor\log k\rfloor}}N(n)\geq\frac{B_{k}^{\lfloor\log
k\rfloor}}{\lfloor\log k\rfloor M_{k}}=e^{k\log^{2}k-(1+o(1))k\log k\log\log
k}$

Let $n_{k}$ be the integer, where the maximum is taken. Note that

$n_{k}<P_{k\lfloor\log
k\rfloor}<P_{(k+1)\lfloor\log(k+1)\rfloor}-\lfloor\log(k+1)\rfloor
M_{k+1}<n_{k+1},$

if $k$ is sufficiently large. Thus the set $\\{n_{k}:k\geq 2\\}$ is infinite.
We have also

$\log n_{k}=(1+o(1))\log P_{k\lfloor\log k\rfloor}=(1+o(1))k\log^{2}k,$
$\log\log n_{k}=(1+o(1))\log k,$

and

$\log\log\log n_{k}=(1+o(1))\log\log k.$

Hence

$\displaystyle N(n_{k})$ $\displaystyle\geq e^{k\log^{2}k-(1+o(1))k\log
k\log\log k}$ $\displaystyle\geq P_{k\lfloor\log k\rfloor}e^{-(3+o(1))k\log
k\log\log k}$ $\displaystyle\geq n_{k}e^{-(3-o(1))\frac{\log n_{k}\log\log\log
n_{k}}{\log\log n_{k}}}.$

Finally we prove 3.10. For a positive integer $k$, let

$\\{S_{1}^{(1)},\ldots,S_{b_{1}}^{(1)}\\},\,\\{S_{1}^{(2)},\ldots,S_{b_{2}}^{(2)}\\},\ldots,\,\\{S_{1}^{(B_{k})},\ldots,S_{b_{B_{k}}}^{(B_{k})}\\}$

be the partitions of $\\{1,...,k\\}.$
For a given $(i_{1},\ldots,i_{\lfloor\log
k\rfloor})\in\\{1,\ldots,B_{k}\\}^{\lfloor\log k\rfloor}$ and any $1\leq
j\leq\lfloor\log k\rfloor$, let

$x_{b_{i_{1}}+\ldots+b_{i_{j-1}}+v}=\prod_{s\in S_{v}^{(i_{j})}}p_{(j-1)k+s},$

where $1\leq v\leq a_{i_{j}}$. For any $1\leq j\leq\lfloor\log k\rfloor$,

$\prod_{1\leq v\leq
b_{i_{j}}}x_{b_{i_{1}}+\ldots+b_{i_{j-1}}+v}=\frac{P_{jk}}{P_{(j-1)k}}$

and

$\sum_{1\leq v\leq b_{i_{1}}+\ldots+b_{i_{\lfloor\log k\rfloor}}}x_{v}\leq
M_{k}\lfloor\log k\rfloor.$

We complete the sequence $(x_{1},\ldots,x_{b_{1}+\ldots+b_{\lfloor\log
k\rfloor}})$ with 1s in order to obtain a solution to equation (1.1) for some
positive integer $n$. Thus,

$P_{k\lfloor\log k\rfloor}=x_{1}\cdot\ldots\cdot
x_{b_{i_{1}}+\ldots+b_{i_{\lfloor\log k\rfloor}}}\cdot 1\cdot\ldots\cdot
1=x_{1}+\ldots+x_{b_{i_{1}}+\ldots+b_{i_{\lfloor\log k\rfloor}}}+1+\ldots+1,$

where the number of summands on the right hand side is $n$. It follows that

(3.12) $\begin{split}P_{k\lfloor\log k\rfloor}&\geq
n=b_{i_{1}}+\dots+b_{i_{\lfloor\log k\rfloor}}+P_{k\lfloor\log
k\rfloor}-(x_{1}+\ldots+x_{b_{i_{1}}+\ldots+b_{i_{\lfloor\log k\rfloor}}})\\\
&>P_{k\lfloor\log k\rfloor}-M_{k}\lfloor\log k\rfloor,\end{split}$

Note that, if

$(i_{1},\ldots,i_{\lfloor\log
k\rfloor})\neq(i_{1}^{\prime},\ldots,i_{\lfloor\log k\rfloor}^{\prime}),$

then

$(x_{1},\ldots,x_{b_{i_{1}}+\ldots b_{i_{\lfloor\log
k\rfloor}}})\neq(x_{1}^{\prime},\ldots,x_{b_{i_{1}^{\prime}}+\ldots
b_{i_{\lfloor\log k\rfloor}^{\prime}}}^{\prime}),$

so different sequences $(i_{1},\ldots,i_{\lfloor\log
k\rfloor})\in\\{1,\ldots,B_{k}\\}^{\lfloor\log k\rfloor}$ yield different
solutions to equation (1.1), where $n$ is bounded by inequality (3.12), which
proves the lower bound (3.10). ∎

## Acknowledgements.

The research was supported by the Lendület program of the Hungarian Academy of
Sciences (MTA). Csaba Sándor was also supported by the National Research,
Development and Innovation Office NKFIH (Grant Nr. K146387) and grant NKFI
KKP144059 ”Fractal ´ geometry and applications.

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