# Abel universal functions: boundary behaviour and Taylor polynomials

Stéphane Charpentier, Myrto Manolaki, Konstantinos Maronikolakis  Stéphane
Charpentier, Institut de Mathématiques, UMR 7373, Aix-Marseille Universite, 39
rue F. Joliot Curie, 13453 Marseille Cedex 13, France
<EMAIL_ADDRESS>Myrto Manolaki, UCD School of Mathematics
and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
<EMAIL_ADDRESS>Konstantinos Maronikolakis, UCD School of Mathematics and
Statistics, University College Dublin, Belfield, Dublin 4, Ireland
<EMAIL_ADDRESS>
###### Abstract.

A holomorphic function $f$ on the unit disc ${\mathbb{D}}$ belongs to the
class $\mathcal{U}_{A}(\mathbb{D})$ of Abel universal functions if the family
$\\{f_{r}:0\leq r<1\\}$ of its dilates $f_{r}(z):=f(rz)$ is dense in the space
of continuous functions on $K$, for any proper compact subset $K$ of the unit
circle. It has been recently shown that $\mathcal{U}_{A}(\mathbb{D})$ is a
dense $G_{\delta}$ subset of the space of holomorphic functions on
$\mathbb{D}$ endowed with the topology of local uniform convergence. In this
paper, we develop further the theory of universal radial approximation by
investigating the boundary behaviour of functions in
$\mathcal{U}_{A}(\mathbb{D})$ (local growth, existence of Picard points and
asymptotic values) and the convergence properties of their Taylor polynomials
outside $\mathbb{D}$.

###### Key words and phrases:

universal radial approximation, dilation, boundary behaviour, universal Taylor
series

###### 2020 Mathematics Subject Classification:

30K15, 30B30, 30E10

The research conducted in this paper was funded by the Irish Research Council
and the French Ministry of Foreign Affairs under the Ulysses 2022 Scheme (No
48450RA). Konstantinos Maronikolakis acknowledges financial support from the
Irish Research Council through the grant [GOIPG/2020/1562].

##  1\. Introduction

Let $H(\mathbb{D})$ be the space of holomorphic functions on the unit disc
$\mathbb{D}$ endowed with the topology of local uniform convergence. It is
known that most functions in $H(\mathbb{D})$ have maximal cluster sets along
any radius. More specifically, Kierst and Szpilrajn in [32] showed that the
set of all functions $f$ in $H(\mathbb{D})$ with the property that, for any
$\zeta$ in the unit circle $\mathbb{T}$ and any $w\in\mathbb{C}$, there exists
a sequence $(r_{n})$ in $[0,1)$ converging to $1$ such that
$\lim_{n\to\infty}f(r_{n}\zeta)=w$, is residual in $H(\mathbb{D})$ (that is,
it contains a dense $G_{\delta}$ set). An analogue of this result holds if we
replace radii by paths with endpoints on the unit circle. The existence of
such functions was established (among more general results) in [10] and the
residuality in [8] (see also [9, 6, 7] for properties of such functions).
Another variant of the result of Kierst and Szpilrajn was shown by Bayart in
[4], who proved that the set of all functions $f$ in $H(\mathbb{D})$ with the
property that, for any measurable function $\varphi$ on $\mathbb{T}$ there
exists a sequence $(r_{n})$ in $[0,1)$ tending to $1$ such that
$f(r_{n}\zeta)\to\varphi(\zeta)$ as $n\to\infty$ for a.e. $\zeta$ in
$\mathbb{T}$, is residual in $H(\mathbb{D})$.

In this paper, we will focus on a class of functions in $H({\mathbb{D}})$ that
possess an even more chaotic boundary behaviour:

######  Definition (Abel universal function).

A function $f$ in $H({\mathbb{D}})$ is called Abel universal if it satisfies
the following property: for every compact set $K$ in the unit circle
$\mathbb{T}$, different from $\mathbb{T}$, and every continuous function
$\varphi$ on $K$, there exists a sequence $(r_{n})$ in $[0,1)$ tending to $1$
such that $\sup_{\zeta\in K}|f(r_{n}\zeta)-\varphi(\zeta)|\to 0$ as
$n\to\infty$. The class of Abel universal functions will be denoted by
${\mathcal{U}}_{A}({\mathbb{D}})$.

In [12], it was shown that $\mathcal{U}_{A}(\mathbb{D})$ is a dense
$G_{\delta}$ subset of $H(\mathbb{D})$, which unifies the results in [32],
[10], [8] and [4] that we mentioned before. In higher dimensions the situation
is more delicate. Indeed, in [27], it was shown that no holomorphic function
on the unit ball $B$ of $\mathbb{C}^{N}$ ($N\geq 2$) can be unbounded (and
hence dense) along every path with endpoint on $\partial B$. More recently,
Globevnik, to address a conjecture in complex geometry from 1977, constructed
a holomorphic function on $B$ which is unbounded along any path of finite
length with endpoint on $\partial B$ (see [26]). It turns out that, if we
restrict to paths of finite length, we can replace unbounded by dense and get
an analogue of the result in [8] that we mentioned before for several complex
variables (see [13]).

There is a growing literature about properties and variants of Abel universal
functions [12, 13, 16, 34, 38, 14], but there is still much scope for
investigation. Our objective is to study two different, but intimately
connected, aspects of Abel universal functions: i) their behaviour as we
approach the boundary through certain regions in $\mathbb{D}$, and ii) the
behaviour of their Taylor polynomials outside $\mathbb{D}$.

It is easy to see that a function $f$ in $H({\mathbb{D}})$ is Abel universal
if and only if the family $\\{f_{r}:\,0\leq r<1\\}$ of its dilates
$f_{r}(z):=f(rz)$, $z\in\overline{{\mathbb{D}}}$, is dense in the space $C(K)$
of complex-valued continuous functions on $K$, for any compact subset $K$ of
${\mathbb{T}}$, different from ${\mathbb{T}}$. Dilation is one of the most
standard techniques used in the study of the boundary behaviour of functions
in $H({\mathbb{D}})$ since it enables us to move from the unknown territory of
the boundary to a region in $\mathbb{D}$, where our function is well defined
and nicely behaved. For a comprehensive overview of dilation theory over the
past century we refer the reader to the recent survey of Mashreghi [35]. The
other standard technique for studying the boundary behaviour of functions in
$H({\mathbb{D}})$ involves considering the partial sums of their Taylor
expansion, which also behave nicely in $\mathbb{D}$ since they converge
locally uniformly to our function. Thus, the class
$\mathcal{U}_{A}(\mathbb{D})$ of Abel universal functions is the natural
analogue of the well-studied class $\mathcal{U}_{T}(\mathbb{D})$ of universal
Taylor series. We say that a function $f$ in $H(\mathbb{D})$ belongs to
$\mathcal{U}_{T}(\mathbb{D})$ if, for every compact set
$K\subset{\mathbb{C}}\setminus\mathbb{D}$ with connected complement, the set
$\\{S_{n}(f):n\in\mathbb{N}\\}$ of the partial sums of its Taylor expansion
about $0$, is dense in the space of functions in $C(K)$ that are holomorphic
in the interior of $K$. In 1996, Nestoridis showed that
$\mathcal{U}_{T}(\mathbb{D})$ is a dense $G_{\delta}$ subset of
$H(\mathbb{D})$ [42]. Since then, various properties of universal Taylor
series have been intensively studied, such as growth [36, 37, 21], gap
structure of the Taylor expansion [25, 40], and boundary behaviour [18, 21,
19, 23]. For a detailed account of several interesting boundary properties of
functions in $\mathcal{U}_{T}(\mathbb{D})$ and the significance of the role of
potential theory in their proofs, we refer to the survey of Gardiner [20]. We
mention below three motivating results, which show that universal Taylor
series have an extremely wild boundary behaviour. Let
$f\in\mathcal{U}_{T}(\mathbb{D})$. Then:

  * (a)

$f$ satisfies a Picard-type property near each boundary point; that is, for
every region of the form $D_{\zeta,r}:=\\{z\in\mathbb{D}:|z-\zeta|<r\\}$,
where $r>0$ and $\zeta\in{\mathbb{T}}$, the image $f(D_{\zeta,r})$ is the
entire complex plane $\mathbb{C}$ except possibly a point [21, Corollary 2].

  * (b)

For a.e. $\zeta$ in ${\mathbb{T}}$ and any open triangle $T_{\zeta}$ in
${\mathbb{D}}$ which has vertex at $\zeta$ and is symmetric about the radius
$[0,\zeta]$, the image $f(T_{\zeta})$ is dense in $\mathbb{C}$. In particular,
$f$ cannot have nontangential limits at a boundary set of positive measure
[19, Theorem 2].

  * (c)

There is a residual subset $Z$ of ${\mathbb{T}}$ such that the image
$\\{f(r\zeta):0\leq r<1\\}$ is dense in $\mathbb{C}$ for every $\zeta\in Z$
[19, Corollary 3].

We note that in part (c) (which follows from part (b)), the set $Z$ cannot be
replaced by ${\mathbb{T}}$. Indeed, in [17] it is shown that given any closed
and nowhere dense subset $E$ of ${\mathbb{T}}$, there is a function in
$\mathcal{U}_{T}(\mathbb{D})$ that has finite radial limits on $E$. This shows
that $\mathcal{U}_{T}(\mathbb{D})$ is not contained in
$\mathcal{U}_{A}(\mathbb{D})$. The converse is also true, since all functions
in $\mathcal{U}_{T}(\mathbb{D})$ have Ostrowski gaps [25, 40], which is not
the case for all functions in $\mathcal{U}_{A}(\mathbb{D})$ [12]. It is worth
mentioning that the existence of Ostrowski gaps played a significant role in
showing that universal Taylor series have the Picard-type property (a). Thus,
it is natural to ask if this property holds for Abel universal functions.

In Section 3, we show that the classes $\mathcal{U}_{T}(\mathbb{D})$ and
$\mathcal{U}_{A}(\mathbb{D})$, although not strongly correlated, have similar
boundary behaviour. In particular, we prove that such functions can have
arbitrary growth on each closed set $A\subset{\mathbb{D}}$ such that
$\overline{A}\cap{\mathbb{T}}$ is a singleton (see Theorem 3.9). We also show
that this result, which is the dilation-analogue of [19, Proposition 6], fails
if $\overline{A}$ contains two paths in ${\mathbb{D}}$ with two different
endpoints on $\mathbb{T}$ (see Proposition 3.12). As an application, we deduce
that all functions in $\mathcal{U}_{A}(\mathbb{D})$ satisfy the Picard-type
property (a) near each boundary point. Interestingly, we note that this
property, as well as many other boundary properties of Abel universal
functions (see Corollary 3.6), can be deduced using results about Valiron
functions [3] and the MacLane class [33, 28]. For example, one such boundary
property is that all functions in $\mathcal{U}_{A}(\mathbb{D})$ admit $\infty$
as an asymptotic value. Finally, we discuss their angular boundary behaviour
(see Corollary 3.8) and conclude the section with some open questions.

As we saw before, the chaotic behaviour of the Taylor polynomials of functions
in $\mathcal{U}_{T}(\mathbb{D})$ outside $\mathbb{D}$ endows them with an
extremely wild boundary behaviour. In Section 4 we study the dual question:
does the wild boundary behaviour of functions in $\mathcal{U}_{A}(\mathbb{D})$
cause a chaotic behaviour of their Taylor polynomials outside the disc of
convergence? Since the intersection
$\mathcal{U}_{T}(\mathbb{D})\cap\mathcal{U}_{A}(\mathbb{D})$ is residual in
$H(\mathbb{D})$, it is tempting to think that the partial sums $S_{n}(f)$ of
the Taylor expansion of a function $f$ in $\mathcal{U}_{A}(\mathbb{D})$ will
have a similar behaviour outside $\mathbb{D}$ (or at least on ${\mathbb{T}}$)
with the ones of functions in $\mathcal{U}_{T}(\mathbb{D})$; that is, for each
$\zeta$ in $\mathbb{C}\setminus{\mathbb{D}}$ (or in ${\mathbb{T}}$), the set
$\\{S_{n}(f)(\zeta):n\in\mathbb{N}\\}$ will be dense in $\mathbb{C}$. We note
that if $f$ is a function in $H({\mathbb{D}})$ which has a maximal cluster set
along the radius through some point $\zeta\in{\mathbb{T}}$, we can apply the
formula

$f(r\zeta)=\sum_{k=0}^{\infty}r^{k}(1-r)S_{k}(f)(\zeta)\ \ \ (r\in[0,1))$

to deduce that the set $\\{S_{n}(f)(\zeta):n\in\mathbb{N}\\}$ is unbounded. It
turns out, that for certain Abel universal functions we cannot replace
unbounded by dense. Indeed, as it was shown in [16], there are Abel universal
functions $f$ such that $\\{S_{n}(f)(1):n\in\mathbb{N}\\}$ is not dense. In
fact, we will prove the following much stronger result:

######  Theorem 1.1 (Theorem 4.13, Corollary 4.8, Corollary 4.12).

  *   * (a)

Given any countable set $E$ in $\mathbb{T}$, there exists
$f\in\mathcal{U}_{A}({\mathbb{D}})$ such that $S_{n}(f)(\zeta)\to\infty$ as
$n\to\infty$ for each $\zeta\in E$.

  * (b)

There exists $f\in\mathcal{U}_{A}({\mathbb{D}})$ such that
$S_{n}(f)(\zeta)\to\infty$ as $n\to\infty$ for a.e.
$\zeta\in\partial\mathbb{D}$ (with respect to the arc length measure) and
$S_{n}(f)(\zeta)\to\infty$ as $n\to\infty$ for a.e.
$\zeta\in{\mathbb{C}}\setminus\overline{{\mathbb{D}}}$ (with respect to the
2-dimensional Lebesgue measure).

  * (c)

There exists $f\in\mathcal{U}_{A}({\mathbb{D}})$ such that $(S_{n}(f))_{n}$
converges to $\infty$ locally in capacity on
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$; that is, for every compact set
$K$ in ${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$ and $M>0$, we have

$\lim_{n\to\infty}\operatorname{cap}\left(\left\\{z\in K:|S_{n}(f)(z)|\leq
M\right\\}\right)=0.$

We note that the set of all such functions is of first Baire category and the
proofs of (a) and (b) are constructive. For part (c), we show that there
exists $f\in\mathcal{U}_{A}({\mathbb{D}})$ without Hadamard-Ostrowski gaps,
which improves the result in [12] about Ostrowski gaps.

Finally, in Section 5, we discuss similar questions about the functions
mentioned in the first paragraph (see Definition 5.1). In addition, we propose
some further developments in connection with classical function spaces.

##  2\. Notation and main definitions

In this section we introduce the notation and main definitions that will be
frequently used throughout this paper. Let us start with the standard ones.
First, we will denote by ${\mathbb{N}}:=\\{0,1,2,\ldots\\}$ the set of non-
negative integers. If $P$ is a polynomial, then $\text{deg}(P)$ will stand for
the degree of $P$. The disc and the circle of centre $a$ and radius $r$ will
be respectively denoted by $D(a,r)$ and $C(a,r)$. If $f$ is a function defined
on ${\mathbb{D}}$ and $r\in[0,1)$, then $f_{r}$ will denote the dilate of $f$
defined by $f_{r}(z)=f(rz)$, $z\in\overline{{\mathbb{D}}}$. Also, given any
$r\in\mathbb{R}$ and any set $A$ in $\mathbb{C}$, we write $rA=\\{rz:z\in
A\\}$. For a power series $f(z)=\sum_{k}a_{k}z^{k}$ and $n\in{\mathbb{N}}$,
the notation $S_{n}(f)$ will be used to denote the $n$-th partial sums of $f$;
that is $S_{n}(f)(z):=\sum_{k=0}^{n}a_{k}z^{k},z\in{\mathbb{C}}$.

We will use the notation $m$ to denote the arclength measure on the unit
circle $\mathbb{T}$ and $\lambda$ for the Lebesgue measure in ${\mathbb{C}}$
(identified with $\mathbb{R}^{2}$). For a compact set $K$ in $\mathbb{C}$, we
denote by $C(K)$ the Banach space consisting of all functions continuous on
$K$, endowed with the supremum norm $\|\cdot\|_{K}$. The notation $A(K)$ will
be used for the algebra of $K$; that is the set of all functions in $C(K)$
that are holomorphic in the interior $K^{\circ}$ of $K$.

Let us recall the definition of Abel universal functions. When not
specifically mentioned, the letter $\rho$ will always stand for an increasing
sequence $(r_{n})_{n}$ in $[0,1)$ such that $r_{n}\to 1$.

######  Definition 2.1.

Let $\rho$ be as above. We denote by ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$
the set of all functions $f\in H({\mathbb{D}})$ that satisfy the following
property: for any compact set $K\subset{\mathbb{T}}$, different from
${\mathbb{T}}$, and any function $\varphi\in C(K)$, there exists an increasing
sequence $(n_{k})$ in ${\mathbb{N}}$ such that

$\|f_{r_{n_{k}}}-\varphi\|_{K}\to 0,\quad k\to\infty.$

We recall that for any $\rho$ the set ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$
is a dense $G_{\delta}$ subset of $H({\mathbb{D}})$ [12]. Moreover, it is
easily seen that the set

${\mathcal{U}}_{A}({\mathbb{D}}):=\bigcup_{\rho}{\mathcal{U}}_{A}({\mathbb{D}},\rho),$

where the union runs over all increasing sequences $\rho\in[0,1)$ that
converges to $1$, coincides with the set of functions $f\in H({\mathbb{D}})$
that satisfy the following property: for any $\varepsilon>0$, any compact set
$K\subset{\mathbb{T}}$, different from ${\mathbb{T}}$, and any function
$\varphi\in C(K)$, there exists $r\in[0,1)$ such that

$\|f_{r}-\varphi\|_{K}\leq\varepsilon.$

As in [16], the terminology _Abel universal function_ will refer to the
elements of ${\mathcal{U}}_{A}({\mathbb{D}})$.

Finally, let us set some technical notations that will be repeatedly used in
several proofs.

  * •

$(\varepsilon_{n})_{n}$ will be a sequence of positive real numbers decreasing
to $0$ with $\sum_{n=1}^{\infty}\varepsilon_{n}\leq 1/2$;

  * •

$(\varphi_{n})_{n}$ will be an enumeration of the polynomials with
coefficients in ${\mathbb{Q}}+i{\mathbb{Q}}$;

  * •

$(K_{n})_{n}$ will be a sequence of proper compact arcs in ${\mathbb{T}}$,
such that for any compact set $K\subset{\mathbb{T}}$, different from
${\mathbb{T}}$, there exists $n\in{\mathbb{N}}$ such that $K\subset K_{n}$;

  * •

$\alpha,\beta:{\mathbb{N}}\to{\mathbb{N}}$ will be two functions such that for
any $l,m\in{\mathbb{N}}$, there exists infinitely many $n\in{\mathbb{N}}$ for
which $(\alpha(n),\beta(n))=(l,m)$.

##  3\. Growth and boundary behaviour of Abel universal functions

The purpose of this section is to exhibit properties of Abel universal
functions in terms of notions that are classical in function theory: the
maximum modulus function, asymptotic values, normality and Picard’s property.
Our first result will be derived from known results that we shall briefly
present together with the necessary definitions.

The maximum modulus function $M(r,f):=\max\\{|f(z)|:\,|z|=r\\}$, $r\in[0,1)$
and $f\in H({\mathbb{D}})$, is commonly used to describe the growth of
holomorphic functions. The next definition is that of asymptotic value, as it
was introduced by Valiron [47], see also [33].

######  Definition 3.1.

A function $f\in H({\mathbb{D}})$ admits an asymptotic value
$\alpha\in{\mathbb{C}}\cup\\{\infty\\}$ if there exists a path
$\gamma:[0,1)\to{\mathbb{D}}$ with $|\gamma(r)|\to 1$ as $r\to 1$ such that
$f\circ\gamma(r)\to\alpha$ as $r\to 1$.

We say that $f$ admits an asymptotic value
$\alpha\in{\mathbb{C}}\cup\\{\infty\\}$ at a point $\zeta\in{\mathbb{T}}$ if
there exists a path $\gamma:[0,1)\to{\mathbb{D}}$ with $\gamma(r)\to\zeta$ as
$r\to 1$ such that $f\circ\gamma(r)\to\alpha$ as $r\to 1$.

Interestingly, the existence of asymptotic values for unbounded holomorphic
functions can be related to their behaviour along spirals, where by spiral we
mean a subset of ${\mathbb{D}}$ of the form
$\\{r(t)e^{i\theta(t)}:\,t\in(0,+\infty)\\}$, with $r(t)\to 1$ and
$\theta(t)\to+\infty$ or $-\infty$ as $t\to+\infty$. Indeed, a result by
Valiron asserts that an unbounded holomorphic function in ${\mathbb{D}}$ that
is bounded on a spiral must admit $\infty$ as an asymptotic value [48]. These
functions, that we will refer to as Valiron functions, like in [3], have been
thoroughly studied, for example in [3, 39, 49]. Note that by [33, p. 71],
there exist functions in $H({\mathbb{D}})$ that are unbounded near each point
of ${\mathbb{T}}$ but that do not admit $\infty$ as an asymptotic value. By
Valiron’s result, they are of course bounded on no spirals in ${\mathbb{D}}$.

A link between the rate of growth of the maximum modulus function and the
asymptotic values of functions holomorphic in ${\mathbb{D}}$ can be made in
studying the MacLane class ${\mathcal{A}}$ of all functions in
$H({\mathbb{D}})$ admitting an asymptotic value at any point of a dense subset
of ${\mathbb{T}}$ [33]. This link is specified by the following result:

######  Theorem 3.2 (Hornblower, [28]).

If $f\in H({\mathbb{D}})$ satisfies the condition

$\int_{0}^{1}\log^{+}\log^{+}M(r,f)dr<\infty,$

then $f$ belongs to ${\mathcal{A}}$.

It is not difficult to see that Valiron functions cannot belong to the MacLane
class ${\mathcal{A}}$ [3]. Therefore, if $f\in H({\mathbb{D}})$ is a Valiron
function, then

$\int_{0}^{1}\log^{+}\log^{+}M(r,f)dr=\infty.$

To state our results we need to introduce some definitions.

######  Definition 3.3.

Let $f$ be holomorphic in ${\mathbb{D}}$. Then $f$ is said to be normal if

$\sup_{z\in{\mathbb{D}}}(1-|z|^{2})f^{\\#}(z)<\infty,$

where $f^{\\#}$ denotes the spherical derivative of $f$, defined by
$f^{\\#}(z)=\dfrac{|f^{\prime}(z)|}{1+|f(z)|^{2}}$.

The term _normal_ comes from Marty’s normality criterion [1, Theorem 17, p.
226] that asserts that $f\in H({\mathbb{D}})$ is normal if and only if the
family $\\{f\circ\varphi:\,\varphi\text{ automorphism of }{\mathbb{D}}\text{
onto itself}\\}$ is normal. We refer to [44, Section 9.1] for some classical
results about normal functions. By a conformal mapping, this definition can be
extended to any simply connected domain. More generally, we will say that a
function $f\in H({\mathbb{D}})$ is normal at no boundary point
$\zeta\in{\mathbb{T}}$ if $f$ is normal in no domain of the form
$D(\zeta,r)\cap{\mathbb{D}}$, $0<r<1$, $\zeta\in{\mathbb{T}}$; that is

$\sup_{D(\zeta,r)\cap{\mathbb{D}}}(1-|z|^{2})f^{\\#}(z)=\infty,\quad
0<r<1,\,\zeta\in{\mathbb{T}}.$

By Montel’s theorem, a holomorphic function in a domain $D$ that omits two
values in ${\mathbb{C}}$, is normal in $D$. Therefore, a function $f$ that is
normal at no boundary point $\zeta\in{\mathbb{T}}$ will omit at most one value
in ${\mathbb{C}}$ in each set of the form $D(\zeta,r)\cap{\mathbb{D}}$,
$0<r<1$. This leads to the following definition:

######  Definition 3.4.

Let $f$ be a holomorphic function in $\mathbb{D}$ and $\zeta$ in $\mathbb{T}$.
We say that $\zeta$ is a Picard point of $f$ if, for each $r>0$, the set
$f(D(\zeta,r)\cap\mathbb{D})$ is the whole complex plane $\mathbb{C}$ except
at most one point.

Observe that if $\zeta\in{\mathbb{T}}$ is a Picard point of $f$ then the
following seemingly stronger condition holds: $f$ assumes every complex value
except possibly one infinitely often in $D(\zeta,r)\cap\mathbb{D}$ for every
$r>0$.

In [3] the authors remark that a Valiron function is normal at no point
$\zeta\in{\mathbb{T}}$ and thus that every point in ${\mathbb{T}}$ is a Picard
point of such a function. This also implies that any Valiron function that
does not vanish in ${\mathbb{D}}$ admits $0$ as an asymptotic value, see [3,
p. 817]. The quite nice Theorems 1.2 and 1.3 in [3] allow us to infer that a
large category of unbounded functions in $H({\mathbb{D}})$ enjoys a wild
boundary behaviour with respect to the notions introduced above. For a subset
$E$ of ${\mathbb{D}}$, we denote by $E_{NT}$ the set of all
$\zeta\in{\mathbb{T}}$ with the property that there exists a sequence
$(z_{n})_{n}$ in $E$ with $z_{n}\to\zeta$ nontangentially.

######  Theorem 3.5 (Barth and Rippon, [3]).

Let $f\in H({\mathbb{D}})$ be bounded on a non-empty subset $E$ of
${\mathbb{D}}$. Then:

  1. (a)

if $f$ is unbounded in ${\mathbb{D}}$ and $m({\mathbb{T}}\setminus E_{NT})=0$,
then $f$ admits $\infty$ as an asymptotic value;

  2. (b)

if $f$ is unbounded near each point of ${\mathbb{T}}$ and
${\mathbb{T}}\setminus E_{NT}$ is nowhere dense in ${\mathbb{T}}$, then

    1. (i)

$f$ is normal at no point of ${\mathbb{T}}$;

    2. (ii)

every point of ${\mathbb{T}}$ is a Picard point for $f$.

This theorem is very helpful for our purpose, which is to describe the
boundary behaviour of Abel universal functions. Indeed, it is easily seen, by
the definition, that for any $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, there
exists $E\subset{\mathbb{D}}$ on which $f$ is bounded and such that
$E_{NT}={\mathbb{T}}$. In addition, it is clear that Abel universal functions
cannot belong to the MacLane class ${\mathcal{A}}$. Thus, by combining the
above, we can deduce the following corollary:

######  Corollary 3.6.

Any function $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ satisfies the
following properties:

  1. (a)

$\int_{0}^{1}\log^{+}\log^{+}M(r,f)dr=\infty$;

  2. (b)

$f$ admits $\infty$ as an asymptotic value;

  3. (c)

$f$ is normal at no point of ${\mathbb{T}}$;

  4. (d)

every point of ${\mathbb{T}}$ is a Picard point for $f$.

######  Remark 3.7.

(i) Another proof of part (a) can be derived from ideas originally contained
in [36, 37], see [12, Corollary 2.10].

(ii) Part (d) tells us that for any $\zeta\in\mathbb{T}$ and any $0<r<1$, the
set $f(D(\zeta,r)\cap\mathbb{D})$ is the whole complex plane $\mathbb{C}$
except at most one point. This result is optimal in the sense that for any
$a\in{\mathbb{C}}$ there exists $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$
such that $a\notin f({\mathbb{D}})$. This comes from the following result in
[14]: $e^{f}$ is Abel universal whenever $f$ is, hence there are Abel
universal functions that do not vanish in ${\mathbb{D}}$. It is obvious that,
for any $a\in{\mathbb{C}}$, $f+a$ is Abel universal if $f$ is.

(iii) For any $a\in{\mathbb{C}}$, there exists
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for which $a$ is an asymptotic
value. This is a consequence of the fact that for any function
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ that does not vanish in
${\mathbb{D}}$, and for any $a\in{\mathbb{C}}$, the function $a+1/f$ is always
Abel universal (see [14]).

(iv) Properties (a), (b) and (d) are also valid for the class of universal
Taylor series.

Finally, we can use [24] to show that Abel universal functions have chaotic
angular behaviour. Let us recall that the Stolz angle $S_{\zeta}(\alpha)$ with
vertex $\zeta\in{\mathbb{T}}$ and opening $\alpha>1$ is defined by

$S_{\zeta}(\alpha):=\left\\{z\in{\mathbb{D}}:\,|z-\zeta|<\alpha(1-|z|)\right\\}.$

In the next result we use $\lambda_{3}$ to denote the Lebesgue measure on
${\mathbb{R}}^{3}$.

######  Corollary 3.8.

Let $f\in{\mathcal{U}}_{A}(\mathbb{D},\rho)$. For $m$-a.e. point
$\zeta\in{\mathbb{T}}$ and for every Stolz angle $S=S_{\zeta}(\alpha)$ with
vertex $\zeta$,

  * (a)

for $\lambda_{3}$-almost every $(w,r)\in\mathbb{C}\times(0,\infty)$,

$\int_{S\cap f^{-1}(C(w,r))}\left|f^{\prime}(z)\right|\left|dz\right|=\infty;$

  * (b)

for every $(w,r)\in\mathbb{C}\times(0,\infty)$,

$\int_{S\cap f^{-1}(D(w,r))}\left|f^{\prime}\right|^{2}d\lambda=\infty.$

The integral in Corollary 3.8 (a) measures the total arc length of the image
of $S\cap f^{-1}(C(w,r))$ under $f$, taking into account multiplicities. Thus,
although this image is contained in the circle $C(w,r)$, this integral
condition tells us that its length, counting multiplicities, is infinite for
almost every choice of $(w,r)\in\mathbb{C}\times(0,\infty)$. Similarly, the
integral condition in Corollary 3.8 (b) tells us that the total area of the
image of $S\cap f^{-1}(D(w,r))$ under $f$, counting multiplicities, is
infinite for every choice of $(w,r)\in\mathbb{C}\times(0,\infty)$.

###### Proof of Corollary 3.8.

Part (a) follows immediately from combining the fact that Abel universal
functions do not have nontangential limit at any point of $\mathbb{T}$
together with [24, Theorem 1]. Part (b) follows from part (a) and the co-area
formula (see p. 3 in [24]). ∎

It is natural to ask whether an analogue of condition (a) in Corollary 3.6
would hold if we restrict to a smaller boundary region near a point in
${\mathbb{T}}$. For a set $A\subset{\mathbb{D}}$ such that
$\bar{A}\cap{\mathbb{T}}\neq\emptyset$, let $M_{A}(f,r):=\sup\\{|f(z)|:\,z\in
A,\,|z|=r\\}$ be the maximum modulus function of $f$ restricted to $A$.
Condition (a) in Corollary 3.6 tells us that $M_{{\mathbb{D}}}(f,r)$ grows
rather fast to $\infty$ as $r\to 1$ for any Abel universal function $f$. One
can expect that the smaller $\bar{A}\cap{\mathbb{T}}$ is, the weaker the
constraints are on the rate of growth of $M_{A}(f,r)$. The next two results
give some interesting information in this direction. The first one shows that,
given any set $A\subset{\mathbb{D}}$ such that $\overline{A}\cap{\mathbb{T}}$
is a singleton, there exists an Abel universal function whose maximum modulus
function restricted to $A$ has an arbitrary rate of growth.

######  Theorem 3.9.

Let $w:[0,1)\to[1,\infty)$ be continuous and increasing such that
$w(r)\to\infty$ as $r\to 1$. For any $A\subset{\mathbb{D}}$ with
$\overline{A}\cap{\mathbb{T}}$ containing exactly one point, there exists
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that $|f(z)|\leq w(|z|)$ for
any $z\in A$.

###### Proof.

Let $(\varepsilon_{n})_{n}$, $(\varphi_{n})_{n}$, $(K_{n})$, $\alpha$ and
$\beta$ be as in Section 2. Without loss of generality, we may and shall
assume that $\overline{A}\cap{\mathbb{T}}=\\{1\\}$ and that for any $0\leq
r<1$, the set $r\overline{{\mathbb{D}}}\cup\overline{A}$ is compact and has
connected complement. Then for any closed arc $I$ in ${\mathbb{T}}$, the set
$r\overline{{\mathbb{D}}}\cup\overline{A}\cup I$ is also compact and has
connected complement.

We define by induction a sequence $(P_{n})_{n}$ of polynomials and two
increasing sequences $(u_{n})_{n}$ and $(v_{n})_{n}$ in ${\mathbb{N}}$, as
follows. We set $P_{0}\equiv 0$ and $u_{0}=v_{0}=0$. For the inductive step,
let us assume that $(u_{0},v_{0},P_{0}),\ldots,(u_{n-1},v_{n-1},P_{n-1})$ have
been built. Since $w(r)\to\infty$ as $r\to 1$ and $w\geq 1$, we can choose
$u_{n}>u_{n-1}$ such that

(1)
$\left|z^{u_{n}}\left(\varphi_{\alpha(n)}(1)-\sum_{k=0}^{n-1}P_{k}(1)\right)\right|\leq\varepsilon_{n}w(|z|),\quad
z\in A,$

and

(2)
$\left|z^{u_{n}}\left(\varphi_{\alpha(n)}(1)-\sum_{k=0}^{n-1}P_{k}(1)\right)\right|\leq\varepsilon_{n},\quad
z\in r_{{v_{n-1}}}\overline{{\mathbb{D}}}.$

Then we set

$q_{n}(z):=\left\\{\begin{array}[]{lll}z^{-u_{n}}\left(\varphi_{\alpha(n)}(z)-\sum_{k=0}^{n-1}P_{k}(z)\right)&\text{if
}z\in K_{\beta(n)}\\\ \varphi_{\alpha(n)}(1)-\sum_{k=0}^{n-1}P_{k}(1)&\text{if
}z\in\overline{A}\cup r_{{v_{n-1}}}\overline{{\mathbb{D}}}.\end{array}\right.$

By Tietze’s extension theorem, there exists a continuous extension of $q_{n}$
to $\overline{{\mathbb{D}}}$, which we shall also denote by $q_{n}$. From the
hypothesis on $A$, we can apply Mergelyan’s theorem to choose a polynomial
$P_{n}^{*}$ such that for any $z\in\overline{A}\cup
r_{v_{n-1}}\overline{{\mathbb{D}}}\cup K_{\beta(n)}$,

(3) $\left|P_{n}^{*}(z)-q_{n}(z)\right|\leq\varepsilon_{n}.$

By uniform continuity of
$P_{n}^{*}(z)-z^{-u_{n}}\left(\varphi_{\alpha(n)}(z)-\sum_{k=0}^{n-1}P_{k}(z)\right)$,
we may choose $v_{n}>v_{n-1}$ such that for any $z\in K_{\beta(n)}$,

(4)
$\Bigg{|}P_{n}^{*}(r_{v_{n}}z)-(r_{v_{n}}z)^{-u_{n}}\left(\varphi_{\alpha(n)}(r_{v_{n}}z)-\sum_{k=0}^{n-1}P_{k}(r_{v_{n}}z)\right)-\left(P_{n}^{*}(z)-q_{n}(z)\right)\Bigg{|}\leq\varepsilon_{n}.$

Then, we define $P_{n}(z)=z^{u_{n}}P_{n}^{*}(z)$, and set $f:=\sum_{n\geq
0}P_{n}$. Observe that by (1) and (3), for any $n\geq 0$ and any $z\in A$,

$\displaystyle|P_{n}(z)|$ $\displaystyle=$
$\displaystyle\left|z^{u_{n}}P_{n}^{*}(z)\right|$ $\displaystyle\leq$
$\displaystyle\left|z^{u_{n}}\left(P_{n}^{*}(z)-q_{n}(z)\right)\right|+\left|z^{u_{n}}\left(\varphi_{\alpha(n)}(1)-\sum_{k=0}^{n-1}P_{k}(1)\right)\right|$
$\displaystyle\leq$ $\displaystyle\varepsilon_{n}+\varepsilon_{n}w(|z|)$
$\displaystyle\leq$ $\displaystyle 2\varepsilon_{n}w(|z|).$

Similarly, by (2) and (3), we get for any given $k\in\mathbb{N}$ and $n>k$,
and any $|z|\leq r_{v_{k}}$,

(5) $|P_{n}(z)|\leq 2\varepsilon_{n}.$

This shows that $f\in H({\mathbb{D}})$ and that $|f(z)|\leq w(|z|)$ for any
$z\in A$ (we recall that $\sum_{n=1}^{\infty}\varepsilon_{n}\leq 1/2$).

It remains to check that $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Let us
fix $n,m\in{\mathbb{N}}$ and $(l_{k})_{k}$ increasing such that
$\alpha(l_{k})=n$ and $\beta(l_{k})=m$ for any $k\in{\mathbb{N}}$. By (4), (3)
and (5), we get for any $k\in{\mathbb{N}}$ and $z\in K_{m}$,

$\displaystyle\left|f(r_{v_{l_{k}}}z)-\varphi_{n}(r_{v_{l_{k}}}z)\right|$
$\displaystyle=$
$\displaystyle\left|P_{l_{k}}(r_{v_{l_{k}}}z)-\left(\varphi_{n}(r_{v_{l_{k}}}z)-\sum_{i=0}^{l_{k}-1}P_{i}(r_{v_{l_{k}}}z)\right)+\sum_{i\geq
l_{k}+1}P_{i}(r_{v_{l_{k}}}z)\right|$ $\displaystyle\leq$
$\displaystyle\varepsilon_{l_{k}}+|P_{l_{k}}^{*}(z)-q_{l_{k}}(z)|+\sum_{i\geq
l_{k}+1}\left|P_{i}(r_{v_{l_{k}}}z)\right|$ $\displaystyle\leq$ $\displaystyle
2\varepsilon_{l_{k}}+2\sum_{i\geq l_{k}+1}\varepsilon_{i},$

which goes to $0$ as $k\to\infty$. Since
$\varphi_{n}(r_{v_{l_{k}}}z)\to\varphi_{n}(z)$ as $k\to\infty$ uniformly on
$K_{m}$, we deduce that $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

Applying Theorem 3.9 with $A=(0,1)$ and $w(r)=(1-r)^{-1/2},r\in(0,1)$, we get
the existence of $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that

$|f(r)|\leq\frac{1}{\sqrt{1-r}},\quad r\in(0,1).$

This implies $(1-r)|f(r)|\leq(1-r)^{1/2},r\in(0,1)$, hence $(1-r)f(r)\to 0$ as
$r\to 1$ and then the function $z\to(1-z)f(z)$ does not belong to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. We note that for the same function no
antiderivative is Abel universal. This implies the following:

######  Corollary 3.10.

The set ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is not closed under
multiplication by polynomials or under antiderivation.

######  Remark 3.11.

(i) Theorem 3.9 and Corollary 3.10 hold for universal Taylor series as well,
see [19, Proposition 6].

(ii) We also mention that it is proven in [16] that
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is not closed under derivation either.
The corresponding question remains open for universal Taylor series.

Our next result shows that Theorem 3.9 fails whenever $A$ contains two paths
with two different endpoints in ${\mathbb{T}}$.

######  Proposition 3.12.

Let $\gamma_{1},\gamma_{2}:[0,1)\to{\mathbb{D}}$ be two disjoint paths with
$\gamma_{1}(r)\to\zeta_{1}$ and $\gamma_{2}(r)\to\zeta_{2}\in\mathbb{T}$ as
$r\to 1$, for some $\zeta_{1}\neq\zeta_{2}$ in ${\mathbb{T}}$. Let also $h$ be
a positive harmonic function on ${\mathbb{D}}$. Then there is no
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that $|f|\leq e^{h}$ on
$\gamma_{1}\cup\gamma_{2}$.

###### Proof.

It is not difficult to see that it suffices to prove the proposition when
$\gamma_{1}$ and $\gamma_{2}$ are radii, _i.e._
$\gamma_{i}=\left\\{r\zeta_{i}:\,0\leq r<1\right\\}$, $i=1,2$. Let us then
assume that we are in this situation. For the sake of contradiction, suppose
that there exists $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that
$|f(r\zeta_{i})|\leq e^{h(r\zeta_{i})}$ for any $0<r<1$, $i=1,2$. Let $I$
denote a closed subarc of ${\mathbb{T}}$ having $\zeta_{1}$ and $\zeta_{2}$ as
its endpoints. Since $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, there exists
an increasing sequence $(\lambda_{n})_{n}$ of integers such that
$|f(r_{\lambda_{n}}\zeta)|\leq 1$. Hence $\log|f(r_{\lambda_{n}}\zeta)|\leq 0$
for every $\zeta\in I$ and $n\in\mathbb{N}$. Thus, since $h$ is positive, the
function $s:=\log|f|-h$ is negative on $r_{\lambda_{n}}I$ for any
$n\in{\mathbb{N}}$ and by assumption it is also non-positive on
$\gamma_{1}\cup\gamma_{2}$. Note that $s$ is subharmonic in ${\mathbb{D}}$, so
by the maximum principle, $s\leq 0$ on $\left\\{r\zeta:0\leq r\leq
r_{\lambda_{n}}\text{ and }\zeta\in I\right\\}$. Since $r_{\lambda_{n}}\to 1$
as $n\to\infty$, we get that $s\leq 0$ and thus $|f|\leq e^{h}$ on the sector
$\left\\{r\zeta:0\leq r<1\text{ and }\zeta\in I\right\\}$. But, by Fatou’s
Theorem, positive harmonic functions on ${\mathbb{D}}$ have (finite) radial
limits almost everywhere on ${\mathbb{T}}$, which contradicts the fact that
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

Using the previous proposition, we can give an alternative proof of the local
Picard-type property of Abel universal functions.

###### An alternative proof of Corollary 3.6 (d).

For the sake of contradiction, assume that there exists a function
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ that omits two values on
$D(\zeta,r)\cap{\mathbb{D}}$ for some $\zeta\in{\mathbb{T}}$ and $r>0$.
Without loss of generality, we can assume that it omits the values 0 and 1 on
$D(\zeta,r)\cap{\mathbb{D}}$. Using Schottky’s theorem, as in the proof of
[21, Corollary 2], we deduce that $|f(z)|\leq e^{\frac{c}{1-|z|}}$ for $z\in
D(\zeta,r/3)\cap{\mathbb{D}}$, where $c$ is a positive constant. Let

$P(z,\zeta)=\frac{1-|z|^{2}}{|\zeta-z|^{2}},\quad
z\in{\mathbb{D}},\zeta\in{\mathbb{T}},$

denote the Poisson kernel and fix $\zeta_{1},\zeta_{2}\in
D(\zeta,\frac{r}{3})\cap{\mathbb{T}}$, $\zeta_{1}\neq\zeta_{2}$. Then, for
$i=1,2$ and $0\leq s<1$, we have

$P(s\zeta_{i},\zeta_{i})=\frac{1-|s\zeta_{i}|^{2}}{|\zeta_{i}-s\zeta_{i}|^{2}}=\frac{1-s^{2}}{(1-s)^{2}}=\frac{1+s}{1-s}\geq\frac{1}{1-s}.$

We conclude that $P(z,\zeta_{i})\geq(1-|z|)^{-1}$ for
$z\in[0,\zeta_{i}),i=1,2$, and thus, if we set
$h(z)=c(P(z,\zeta_{1})+P(z,\zeta_{2})),z\in{\mathbb{D}}$, we have
$h(z)\geq(1-|z|)^{-1}$ for $z\in[0,\zeta_{1})\cup[0,\zeta_{2})$. Then

$|f(z)|\leq e^{\frac{c}{1-|z|}}\leq e^{h(z)},\quad
z\in([0,\zeta_{1})\cup[0,\zeta_{2}))\cap D(\zeta,r/3).$

Since $h$ is a positive harmonic function on ${\mathbb{D}}$, this contradicts
Proposition 3.12. ∎

We conclude this section by three open questions that naturally arise from the
previous results.

######  Questions 3.13.

(i) Is condition (a) in Corollary 3.6 sharp for Abel universal functions, in
the sense that given any increasing function $w:[0,1)\to[1,\infty)$ that
satisfies $\int_{0}^{1}\log^{+}\log^{+}w(r)dr=\infty$, there exists
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that $M(r,f)\leq w(r)$?

(ii) To prove the local Picard property for universal Taylor series, Gardiner
and Khavinson used their Ostrowski gap structure to show that they have very
strong growth properties at every boundary point (see [21, Theorem 1]). In
particular, they showed that if $\psi:[0,1)\to(0,\infty)$ is an increasing
function such that $\int_{0}^{1}\log^{+}\log^{+}\psi(t)dt<\infty$, and $f$ is
a function in $H(\mathbb{D})$ that satisfies $|f(z)|\leq\psi(|z|)$ on
$D(\zeta,r)\cap\mathbb{D}$ for some $\zeta\in{\mathbb{T}}$ and $r>0$, then
$f\not\in\mathcal{U}_{T}$. Is this true if we replace $\mathcal{U}_{T}$ by
$\mathcal{U}_{A}$?

(iii) Functions satisfying the assumptions of Theorem 3.5 may not be Valiron
functions. Yet, is every Abel universal function a Valiron function?

##  4\. Taylor expansion of Abel universal functions

In this section, we investigate properties of the Taylor polynomials of Abel
universal functions. Specifically, we will focus on their possible gap
structures and the behaviour of their Taylor partial sums outside
${\mathbb{D}}$. Our general aim is to understand to what extent being an Abel
universal function affects the Taylor expansion with respect to these aspects,
and _vice-versa_. Before giving motivations and stating our results, we shall
recall the definitions of three notions of gaps structure.

######  Definition 4.1.

Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ be a power series with radius of
convergence $1$. We say that:

  1. (a)

$f$ possesses _Hadamard gaps_ if there is a sequence $(n_{k})_{k}$ in
${\mathbb{N}}$ with $\inf_{k}n_{k+1}/n_{k}>1$ such that $a_{j}=0$ whenever
$j\neq n_{k}$ for all $k\in\mathbb{N}$;

  2. (b)

$f$ possesses _Hadamard-Ostrowski gaps_ if there are sequences $(p_{k})_{k}$
and $(q_{k})_{k}$ in ${\mathbb{N}}$ with $1\leq p_{1}<q_{1}\leq
p_{2}<q_{2}\leq...$ and $\inf_{k}q_{k}/p_{k}>1$ such that

$\limsup_{k\in I,k\to\infty}|a_{k}|^{\frac{1}{k}}<1,$

where $I=\bigcup_{k=0}^{\infty}\\{p_{k}+1,p_{k}+2,\dots,q_{k}\\}$;

  3. (c)

$f$ possesses _Ostrowski gaps_ if there are sequences $(p_{k})_{k}$ and
$(q_{k})_{k}$ in ${\mathbb{N}}$ with $1\leq p_{1}<q_{1}\leq
p_{2}<q_{2}\leq...$ and $q_{k}/p_{k}\to\infty$ such that

$\limsup_{k\in I,k\to\infty}|a_{k}|^{\frac{1}{k}}=0,$

where $I=\bigcup_{k=0}^{\infty}\\{p_{k}+1,p_{k}+2,\dots,q_{k}\\}$.

It is clear that if $f$ has Ostrowski gaps then it has Hadamard-Ostrowski
gaps. These definitions were motivated by the work of Ostrowski, who showed
that there are intimate connections between the gap structure of a Taylor
series and the behaviour of its partial sums outside the disc of convergence.
It is interesting for our purposes to mention that the existence of Hadamard
gaps is related to the notion of normality, Picard’s property and the MacLane
class, encountered in Section 3. The following theorem, a compilation of
different results, makes this link explicit.

######  Theorem 4.2.

Suppose that $f=\sum_{k}a_{k}z^{k}$ has Hadamard gaps.

  1. (a)

_Hwang[29]_: If $\limsup_{k}|a_{k}|=\infty$, then $f$ assumes every complex
value infinitely often in each sector of the form
$\\{z\in{\mathbb{D}}:\,\alpha<\arg z<\beta\\}$, where $0\leq\alpha<\beta\leq
2\pi$.

  2. (b)

_Sons and Campbell[46]_: If $\limsup_{k}|a_{k}|=\infty$, then $f$ is not
normal in ${\mathbb{D}}$;

  3. (c)

_Murai[41]_: $f$ belongs to the Maclane class ${\mathcal{A}}$.

######  Remark 4.3.

In part (c), the contribution of Murai was to show that if $f$ has Hadamard
gaps and unbounded Taylor coefficients, then $f$ admits the asymptotic value
$\infty$ at _every_ point of ${\mathbb{T}}$. In the case where the
coefficients are bounded, the conclusion follows from an older result of
Paley, see [50].

Assertions (a) and (b) in Theorem 4.2, in comparison with assertions (c) and
(d) in Corollary 3.6, may suggest that Abel universal functions could be found
among Hadamard lacunary series. However, part (c) of Theorem 4.2 rules out
this possibility, since Abel universal functions do not belong to the Maclane
class ${\mathcal{A}}$:

######  Corollary 4.4.

There is no Abel universal function with Hadamard gaps.

In contrast, since the existence of Ostrowski gaps is a more flexible
constraint, it is not difficult to construct an Abel universal function that
has Ostrowski gaps (we refer the reader to [15], where an ad hoc construction
of universal functions in $H({\mathbb{D}})$ with _large_ Ostrowski-type gaps
is displayed). This can also be derived from the fact that every universal
Taylor series possesses Ostrowski gaps [25, 40], the fact that both classes of
Abel universal functions and universal Taylor series are residual in
$H({\mathbb{D}})$, and the Baire category theorem. Nevertheless it is shown in
[12] that there exist Abel universal functions without Ostrowski gaps. This is
not really surprising either since the existence of Ostrowski gaps for
universal Taylor series comes from the approximation properties of their
Taylor partial sums on sets that are non-thin at $\infty$ (see [40] for
details), while Abel universal functions are defined only by properties inside
${\mathbb{D}}$ near points of ${\mathbb{T}}$. This remark is a motivation to
ask whether there are Abel universal functions without Hadamard-Ostrowski
gaps.

Our first result gives an affirmative answer. It even tells us that, given any
prescribed function $g\in H({\mathbb{D}})$, we can find an Abel universal
function whose Taylor coefficients are larger in modulus than those of $g$.

######  Theorem 4.5.

For any sequence $(\gamma_{k})_{k}$ of positive real numbers such that
$\limsup_{k}\gamma_{k}^{1/k}\leq 1$, there exists a function
$f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$
with $|a_{k}|\geq\gamma_{k}$ for every $k$ large enough. In particular, there
exist Abel universal functions without Hadamard-Ostrowski gaps.

For the proof of Theorem 4.5, we will need the following simple lemma, whose
verification is left to the reader.

######  Lemma 4.6.

If $(\gamma_{k})_{k}$ is as in the previous theorem, then for any $0<r<1$, the
series $\sum_{j\geq n}\sum_{k\geq j}\gamma_{k}r^{k}$ tends to $0$ as
$n\to\infty$.

###### Proof of Theorem 4.5.

Let $\rho=(r_{n})_{n}$. Let us fix a sequence $(R_{n})_{n}$ in $[0,1)$ such
that $0<R_{n}<r_{n}<R_{n+1}<r_{n+1}<1$, $n\in{\mathbb{N}}$. We build by
induction sequences $(P_{n})_{n}$ and $(Q_{n})_{n}$ of polynomials and an
increasing sequence $(u_{n})_{n}$ of integers. We set $P_{0}\equiv Q_{0}\equiv
0$ and $u_{0}=0$ and once we have built $P_{0},\ldots,P_{n-1}$,
$Q_{0},\ldots,Q_{n-1}$ and $u_{0},\ldots,u_{n-1}$ we choose
$u_{n}>\deg(P_{n-1})$ and apply Mergelyan’s theorem to get
$P_{n}(z)=\sum_{i\geq u_{n}}a_{i,n}z^{i}$ so that the following hold:

  1. (a)

$\max\left\\{\sum_{i\geq u_{n}}\gamma_{i}r_{n+1}^{i},\sum_{j\geq
u_{n}}\sum_{i\geq
j}\gamma_{i}r_{n}^{i},(r_{n}/R_{n+1})^{u_{n}}/(1-r_{n}/R_{n+1})\right\\}\leq\varepsilon_{n}$;

  2. (b)

$\sup_{|z|\leq R_{n}}|P_{n}(z)|\leq\varepsilon_{n}$;

  3. (c)

$\sup_{z\in
K_{\beta(n)}}|P_{n}(r_{n}z)-\left(\varphi_{\alpha(n)}(z)-\sum_{0\leq j\leq
n-1}(P_{j}+Q_{j})(r_{n}z)\right)|\leq\varepsilon_{n}.$

Note that (a) is possible because of Lemma 4.6. Then we define
$Q_{n}(z)=\sum_{i=\text{deg}(P_{n-1})+1}^{\text{deg}(P_{n})}b_{i,n}z^{i}$ so
that:

  1. (d)

$b_{i,n}=\gamma_{i}$ for any $i=\text{deg}(P_{n-1})+1,\ldots u_{n}-1$;

  2. (e)

for any $i=u_{n},\ldots,\text{deg}(P_{n})$, $b_{i,n}=0$ if
$|\Re(a_{i,n})|\geq\gamma_{i}$ and $b_{i,n}=2\gamma_{i}$ if
$|\Re(a_{i,n})|\leq\gamma_{i}$.

Observe that for any $n$ and any $i$, we have $0\leq b_{i,n}\leq 2\gamma_{i}$
and that, if we set $a_{i,n}=0$ for any $i=\text{deg}(P_{n-1})+1,\ldots
u_{n}-1$, then $|a_{i,n}+b_{i,n}|\geq|\Re(a_{i,n}+b_{i,n})|\geq\gamma_{i}$.
Using (b) and (e), we get that the function $f(z)=\sum_{n\geq
0}(P_{n}+Q_{n})(z)=\sum_{k}a_{k}z^{k}$ is in $H({\mathbb{D}})$ and, by the
previous, that $|a_{k}|\geq\gamma_{k}$ for any $k$.

The proof will be completed once we have proven that
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Let us fix $n,m\in{\mathbb{N}}$
and let $(n_{l})_{l}\subset{\mathbb{N}}$ be such that for any
$l\in{\mathbb{N}}$, $\alpha(n_{l})=n$ and $\beta(n_{l})=m$. By (c), for $z\in
K_{m}$ we have

$\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\leq\varepsilon_{n_{l}}+\left|Q_{n_{l}}(r_{n_{l}}z)\right|+\left|\sum_{j\geq
n_{l}+1}P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right|.$

Moreover it follows from (d), (e) and (a) that

$\left|Q_{n_{l}}(r_{n_{l}}z)\right|\leq\sum_{i\geq
u_{n_{l}-1}}\gamma_{i}r_{n_{l}}^{i}\leq\varepsilon_{n_{l}-1}.$

Now, let us observe that by (b) and Cauchy’s inequalities we have
$\left|a_{i,j}\right|\leq\varepsilon_{j}R_{j}^{-i}$ for any $j\geq n_{l}+1$
and any $i\in{\mathbb{N}}$, whence by (a) and (e), for any $z\in K_{m}$,

$\displaystyle\left|\sum_{j\geq
n_{l}+1}\left(P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right)\right|$
$\displaystyle\leq$ $\displaystyle\sum_{j\geq n_{l}+1}\sum_{i\geq
u_{j}}\varepsilon_{j}\left(\frac{r_{n_{l}}}{R_{j}}\right)^{i}+\sum_{j\geq
n_{l}+1}\sum_{i\geq u_{j-1}}r_{n_{l}}^{i}$ $\displaystyle\leq$
$\displaystyle\sum_{j\geq
n_{l}+1}\varepsilon_{j}\frac{}{}\frac{\left(r_{n_{l}}/R_{n_{l}+1}\right)^{u_{n_{l}}}}{1-r_{n_{l}}/R_{n_{l}+1}}+\sum_{j\geq
u_{n}}\sum_{i\geq j}b_{i}r_{n}^{i}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{j\geq
n_{l}+1}\varepsilon_{j}\right)+\varepsilon_{n_{l}}.$

Altogether we get

$\sup_{z\in K_{m}}\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\rightarrow
0\quad\text{as }l\to 0,$

and so $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

As recalled in the introduction, the partial sums of the Taylor expansion at
$0$ of any Abel universal function must be unbounded at any point in
${\mathbb{T}}$. We will now show that for some Abel universal functions, the
partial sums of their Taylor expansion even go to $\infty$ almost everywhere
outside ${\mathbb{D}}$. This confirms that being Abel universal does not
impose on the Taylor polynomials any oscillatory behaviour (at least on a
large subset of ${\mathbb{C}}\setminus{\mathbb{D}}$).

We shall now see that Theorem 4.5 implies that for some Abel universal
functions $f$, the sequence of partial sums $(S_{n}(f))_{n}$ converges to
$\infty$ _locally in capacity_ in
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$. This will be a consequence of
a result of Kalmes, Müller and Nieß that we will state below. We recall that a
sequence $(f_{n})_{n}$ of Borel-measurable functions on a set
$U\subset{\mathbb{C}}$ converges to $\infty$ in capacity if, for every $M>0$,
we have:

$\lim_{n\to\infty}\text{cap}\left(\left\\{z\in U:|f_{n}(z)|\leq
M\right\\}\right)=0.$

Moreover, if $U$ is open, we say that $(f_{n})_{n}$ converges to $\infty$
_locally_ in capacity if it converges to $\infty$ in capacity on any compact
subset of $U$.

######  Theorem 4.7 (Theorem 1.4, [31]).

Let $f(z)=\sum_{k}a_{k}z^{k}$ be a Taylor series with radius of convergence
$1$ and without Hadamard-Ostrowski gaps. Then $(S_{n}(f))_{n}$ converges to
$\infty$ locally in capacity in
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$.

We deduce from Theorems 4.5 and 4.7 the following result.

######  Corollary 4.8.

There exists $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that
$(S_{n}(f))_{n}$ converges to $\infty$ locally in capacity in
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$. In particular, some
subsequence of $(S_{n}(f))_{n}$ converges pointwise to $\infty$ in
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$, outside a set of capacity $0$.

Furthermore, one can state a measure-theoretical version of the previous
corollary using ideas of Kahane and Melas [30]. Indeed, the proof of [30,
Theorem 2] shows that given any $f\in H({\mathbb{D}})$ there exists a function
$g$ in the Wiener algebra $W$, of power series that are absolutely convergent
in $\overline{{\mathbb{D}}}$, such that $S_{n}(f+g)(z)\to\infty$ for
$\lambda$-a.e. $z\in{\mathbb{C}}\setminus\overline{{\mathbb{D}}}$. Since
$W\subset A({\mathbb{D}})$ and since $f+g$ obviously belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ whenever
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and $g\in A({\mathbb{D}})$, we
immediately get the following complement of Corollary 4.8.

######  Proposition 4.9.

For any $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ there exists $g\in W$ such
that $S_{n}(f+g)\to\infty$ $\lambda$-a.e. in
${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$ while
$f+g\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$.

So far, we have proven that, contrary to what happens for universal Taylor
series, being Abel universal has no effect on the gap structures of its Taylor
coefficients and no influence on the behaviour of its Taylor partial sums on
large subsets of ${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$. It remains to
see if this is still the case on the boundary ${\mathbb{T}}$ of
${\mathbb{D}}$, where the most erratic behaviour of the partial sums could be
expected. The next result gives an affirmative answer. We recall that $m$
stands for the arclength measure on ${\mathbb{T}}$.

The previous show that the influence of the universal property of Abel
universal functions on the possible chaotic behaviour of their partial sums
outside $\overline{{\mathbb{D}}}$ is apparently null. In fact this may be not
be surprising since the universality of Abel universal functions _a priori_
occur only on ${\mathbb{T}}$. The formula
$f(rz)=\sum_{k}a_{k}r^{k}z^{k}=\sum_{k}c_{n,k}S_{k}(f)$ with
$c_{n,k}=r^{k}(1-r)$, valid for any $f\in H({\mathbb{D}})$, any $r\in(0,1)$
and $z\in{\mathbb{T}}$, shows that a wild radial boundary behaviour of $f$ may
have an effect of the behaviour of the partial sums of $f$ on ${\mathbb{T}}$.
Indeed, it implies that if $f\in{\mathcal{U}}_{A}({\mathbb{D}})$, then
$\\{S_{n}(f)(z):\,n\in{\mathbb{N}}\\}$ must be unbounded for any
$z\in{\mathbb{T}}$. The next theorem lets us to think that, in some sense,
this is only effect of Abel universality on a possible wild behaviour of the
partial sums of $f$ on ${\mathbb{T}}$.

######  Theorem 4.10.

There exists $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that
$S_{n}(f)\to\infty$ $m$-a.e. on ${\mathbb{T}}$.

###### Proof.

Let $\rho=(r_{n})_{n}$. We fix a sequence $(R_{n})_{n}$ in $(0,1)$ such that
$R_{n}<r_{n}<R_{n+1}<r_{n+1}$ for any $n\in{\mathbb{N}}$. We will build $f$ by
induction as a sum $\sum_{n}(P_{n}+Q_{n})$ where $P_{n}$ and $Q_{n}$ are two
polynomials that will take the following forms

$P_{n}=\sum_{k=u_{n}}^{\text{deg}(P_{n})}a_{k,n}z^{k},\quad
Q_{n}=\sum_{i=u_{n}}^{\text{deg}(P_{n})}b_{k,n}z^{k},$

where $(u_{n})_{n}$ is an increasing sequence of integers satisfying some
additional growth condition, that will also be defined by induction. At each
step $n$, we first define $u_{n}$, then comes $P_{n}$ and third we choose
$Q_{n}$. To start with, we set $P_{0}\equiv Q_{0}\equiv 0$ and $u_{0}=0$. For
the inductive step, let us assume that
$(u_{0},P_{0},Q_{0}),\ldots,(u_{n-1},P_{n-1},Q_{n-1})$ have been built. We
shall define $u_{n}$, $P_{n}$ and $Q_{n}$ as follows.

We choose $u_{n}\in{\mathbb{N}}$ such that the following conditions are
satisfied.

  1. (a)

$u_{n}\geq\max\\{2,\text{deg}(P_{n-1})+1\\}$;

  2. (b)

$\sum_{j\geq u_{n}}\sum_{k\geq j}k^{4}r_{n}^{k}\leq\varepsilon_{n}$;

  3. (c)

$(r_{n}/R_{n+1})^{u_{n}}\leq 1-r_{n}/R_{n+1}$.

Note that (b) is possible since $\sum_{j\geq n}\sum_{k\geq j}k^{4}r^{k}\to 0$
as $n\to\infty$ for any $r\in(0,1)$ by Lemma 4.6. Then, we apply Mergelyan’s
theorem to define $P_{n}(z)=\sum_{k\geq u_{n}}a_{k,n}z^{k}$ as a polynomial so
that

  1. (d)

$\sup_{|z|\leq R_{n}}|P_{n}(z)|\leq\varepsilon_{n}$;

  2. (e)

$\sup_{z\in
K_{\beta(n)}}|P_{n}(r_{n}z)-\left(\varphi_{\alpha(n)}(z)-\sum_{0\leq j\leq
n-1}(P_{j}+Q_{j})(r_{n}z)\right)|\leq\varepsilon_{n}$

It remains to define $Q_{n}$, which is the most important step of the
construction. To proceed, we will build by a finite induction
$\text{deg}(P_{n})-u_{n}+2$ complex numbers $b_{u_{n}-1,n}$,
$b_{u_{n},n},\ldots,b_{\text{deg}(P_{n}),n}$. First we set $b_{u_{n}-1,n}=0$.
Let us then assume that $b_{u_{n}-1,n}$, $b_{u_{n},n},\ldots,b_{l-1,n}$ have
been chosen for some $l\in\\{u_{n},\ldots,\text{deg}(P_{n})\\}$. To define
$b_{l,n}$ we proceed as follows.

For any complex number $c$, we set

$G_{l}(c)=\left\\{w\in{\mathbb{T}}:\,\left|\sum_{j=0}^{n-1}(P_{j}+Q_{j})(w)+\sum_{k=u_{n}}^{l-1}(a_{k,n}+b_{k,n})w^{k}+a_{l,n}w^{l}+cw^{l}\right|\leq
l\right\\}.$

We observe that $G_{n}(c)\cap G_{n}(c^{\prime})\neq\emptyset$ implies that
$|c-c^{\prime}|\leq 2l$. Observe also that there exists an integer $s_{l}\geq
l^{2}$ and $c_{1},\ldots,c_{s_{l}}$ in the disc $\overline{D(0,l^{4})}$ such
that $|c_{j}-c_{k}|>2l$ for any $j\neq k$ (recall that $l\geq u_{n}\geq 2$).
Therefore the sets $G_{l}(c_{1}),\ldots,G_{l}(c_{s_{l}})$ are pairwise
disjoint closed subsets of ${\mathbb{T}}$, which implies that, for some
$i_{l}\in\\{1,\ldots,c_{s_{l}}\\}$,

$m\left(G_{l}(c_{i_{l}})\right)\leq\frac{m({\mathbb{T}})}{s_{l}}\leq\frac{2\pi}{l^{2}}.$

We set $b_{l,n}=c_{i_{l}}$ to conclude this finite induction and

$Q_{n}=\sum_{k=u_{n}-1}^{\text{deg}(P_{n})}b_{k,n}z^{k}=\sum_{k=u_{n}}^{\text{deg}(P_{n})}b_{k,n}z^{k}$

to conclude the whole induction.

Now we have to check that the power series $f=\sum_{n}(P_{n}+Q_{n})$ satisfies
the desired properties. First note by (d) that the series $\sum_{n}P_{n}$
defines a function in $H({\mathbb{D}})$. Moreover, the construction ensures
that for every $n$ and every $k$ we have $|b_{k,n}|\leq k^{4}$ and so
$\sum_{n}Q_{n}$ also belongs to $H({\mathbb{D}})$. Thus $f$ is well-defined
and belongs to $H({\mathbb{D}})$.

Let us prove that for $m$-a.e. $z\in{\mathbb{T}}$, $S_{N}(f)(z)\to\infty$ as
$N\to\infty$. Set

$I=\cup_{n\geq 1}\\{u_{n},u_{n}+1,\ldots,\text{deg}(P_{n})\\}.$

The construction of the $b_{k,n}$ gives us that for any $k\in I$ and any
$z\in{\mathbb{T}}\setminus G_{k}(b_{k,n})$, we have $|S_{k}(f)(z)|\geq k$,
where $n$ is such that $u_{n}\leq k\leq\text{deg}(P_{n})$. Since
$\sum_{k}m\left(G_{k}(b_{k,n})\right)\leq\sum_{k}\frac{2\pi}{k^{2}}<\infty$,
by the Borel-Cantelli Theorem, $m$-a.e. $z\in{\mathbb{T}}$ belongs to all but
finitely many sets ${\mathbb{T}}\setminus G_{k}(b_{k,n})$. Thus for $m$-a.e.
$z\in{\mathbb{T}}$, $S_{N}(g)(z)\to\infty$ as $N\to\infty$, $N\in I$. Now, for
every $n\geq 1$ and every $N\in\\{\text{deg}(P_{n}),\ldots,u_{n+1}\\}$, we
have $S_{N}(f)(z)=S_{\text{deg}(P_{n})}(f)(z)$ for any $z\in{\mathbb{T}}$.
Therefore $S_{N}(f)(z)\to\infty$ as $N\to\infty$.

To finish the proof, it remains to check that $f$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Let us fix $n,m\in{\mathbb{N}}$ and
let $(n_{l})_{l}\subset{\mathbb{N}}$ be such that for any $l\in{\mathbb{N}}$,
$\alpha(n_{l})=n$ and $\beta(n_{l})=m$. By (e) we have, for any $z\in K_{m}$,

(6)
$\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\leq\varepsilon_{n_{l}}+\left|Q_{n_{l}}(r_{n_{l}}z)\right|+\left|\sum_{j\geq
n_{l}+1}\left(P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right)\right|.$

By the construction of $Q_{n}$ (and $b_{k,n}$) and the choice of $u_{n}$ (see
(b)), we have for any $z\in{\mathbb{T}}$,

$\left|Q_{n_{l}}(r_{n_{l}}z)\right|\leq\sum_{k=u_{n_{l}}}^{\text{deg}(P_{n_{l}})}k^{4}r_{n_{l}}^{k}\leq\varepsilon_{n_{l}}.$

To deal with the third term of the right hand-side of (6), we derive first
from (d) and Cauchy’s inequalities that for any $n\geq 1$ and any
$k\in\\{u_{n},\ldots,\text{deg}(P_{n})\\}$,
$|a_{k,n}|\leq\varepsilon_{n}R_{n}^{-k}$. Then, together with the definition
of the $b_{k,n}$, conditions (b) and (c), we obtain, for any $z\in K_{m}$,

$\displaystyle\left|\sum_{j\geq
n_{l}+1}\left(P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right)\right|$
$\displaystyle\leq$ $\displaystyle\sum_{j\geq n_{l}+1}\sum_{k\geq
u_{j}}\varepsilon_{j}\left(\frac{r_{n_{l}}}{R_{j}}\right)^{k}+\sum_{j\geq
n_{l}+1}\sum_{k\geq u_{j}}k^{4}r_{n_{l}}^{k}$ $\displaystyle\leq$
$\displaystyle\sum_{j\geq
n_{l}+1}\varepsilon_{j}\frac{\left(r_{n_{l}}/R_{n_{l}+1}\right)^{u_{n_{l}}}}{1-r_{n_{l}}/R_{n_{l}+1}}+\sum_{j\geq
u_{n_{l}}}\sum_{k\geq j}k^{4}r_{n_{l}}^{k}$ $\displaystyle\leq$
$\displaystyle\sum_{j\geq n_{l}+1}\varepsilon_{j}+\varepsilon_{n_{l}}.$

Altogether we get

$\sup_{z\in K_{m}}\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\rightarrow
0\quad\text{as }l\to\infty,$

and so $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

######  Remark 4.11.

We note that the previous result allows us to see that Abel’s limit theorem
strongly fails if we replace the assumption “$S_{n}(f)(\zeta)$ converges in
${\mathbb{C}}$” by “$S_{n}(f)(\zeta)\to\infty$”, in the sense that not only
$f(r\zeta)$ may not tend to $\infty$, but $f([0,\zeta))$ can even be dense in
${\mathbb{C}}$.

Combining Proposition 4.9 with Theorem 4.10 we get the following:

######  Corollary 4.12.

There exists an Abel universal function $f$ such that $S_{n}(f)\to\infty$
$\lambda$-a.e. on ${\mathbb{C}}\setminus\overline{{\mathbb{D}}}$ and
$S_{n}(f)\to\infty$ $m$-a.e. on ${\mathbb{T}}$.

In Theorem 4.10, the large set of points in ${\mathbb{T}}$ at which $S_{n}(f)$
tends to $\infty$ is not prescribed. However, notice that given a countable
set $E\in{\mathbb{T}}$ and any set $A$ of full arclength measure in
${\mathbb{T}}$, there exists a rotation $r$ (about $0$) such that $r(E)\subset
A$. Indeed, for any $z\in E$, the set $\\{\zeta\in{\mathbb{T}}:\,\zeta z\in
A\\}$ has clearly full measure so that

$m(\bigcap_{z\in E}\\{\zeta\in{\mathbb{T}}:\,\zeta z\in A\\})=2\pi.$

Thus if we let $r$ be the rotation (multiplication) by any element of
$\cap_{z\in E}\\{\zeta\in{\mathbb{T}}:\,\zeta z\in A\\}$ and let $f$ be an
Abel universal function such that $S_{n}(f)(z)\to\infty$ for every $z$ in some
set $A\subset{\mathbb{T}}$ with full measure, then $g:=f\circ r$ also belongs
to ${\mathcal{U}}_{A}({\mathbb{D}})$ and $S_{n}(g)(z)\to\infty$ for every
$z\in E$. Therefore we have the following consequence of Theorem 4.10.

######  Corollary 4.13.

For any countable set $E\subset{\mathbb{T}}$, there exists
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that $S_{n}(f)(z)\to\infty$
for every $z\in E$.

The proof of this corollary is based on Theorem 4.10 that relies on the Borel-
Cantelli lemma. However one can easily adapt the proof of Theorem 4.10 in
order to get Corollary 4.13 without making use of any probabilistic tool. It
suffices to use instead the following geometric lemma (the details of the
construction of the desired Abel universal function are left to the reader):

######  Lemma 4.14.

Let $w_{1},\dots,w_{l}\in{\mathbb{C}},z_{1},\dots,z_{l}\in{\mathbb{T}}$ and
$R>0$. There exists $b\in\mathbb{C}$ with $|b|\leq 2lR$ such that
$|w_{m}+bz_{m}|\geq R$ for $m=1,\dots,l$.

###### Proof.

Clearly, if $|w_{m}|\geq R$ for $m=1,\dots,l$, then we can select $b=0$ and so
we may assume that at least one of the numbers $w_{1},\dots,w_{l}$ belongs to
$D(0,R)$. Without loss of generality, we assume that $|w_{1}|<R$. Now, if
$|w_{m}|\geq 3R$ for $m=2,\dots,l$, then we can select $b=2R$. Thus, we may
assume that at least one of the numbers $w_{2},\dots,w_{l}$ belongs to
$D(0,3R)$. Continuing in this way, we notice that the worst case is when the
disc $D(0,R)$ and each annulus $\\{z\in{\mathbb{C}}:(2n-1)R\leq|z|<(2n+1)R\\}$
for $n=1,\dots,l-1$ contain exactly one of $w_{1},\dots,w_{l}$. In this case
we can select $b=2lR$. ∎

###### Proof of Theorem 4.13.

We fix a sequence $(R_{n})_{n}$ in $(0,1)$ such that
$R_{n}<r_{n}<R_{n+1}<r_{n+1}$ for any $n$. We will build $f$ by induction as a
sum $f=\sum_{n}(P_{n}+Q_{n})$ where $P_{n}$ and $Q_{n}$ are two polynomials
that will take the following forms

$P_{n}=\sum_{k=u_{n}}^{\text{deg}(P_{n})}a_{k,n}z^{k},\quad
Q_{n}=\sum_{i=u_{n}}^{\text{deg}(P_{n})}b_{k,n}z^{k},$

where $(u_{n})_{n}$ is an increasing sequence of integers satisfying some
additional growth condition, that will also be defined by induction. At each
step $n$, $u_{n}$ is first defined, then comes $P_{n}$ and third we choose
$Q_{n}$. To start with, we set $P_{0}\equiv Q_{0}\equiv 0$ and $u_{0}=0$. For
the inductive step, let us assume that
$(u_{0},P_{0},Q_{0}),\ldots,(u_{n-1},P_{n-1},Q_{n-1})$ have been built. We
shall define $u_{n}$, $P_{n}$ and $Q_{n}$ as follows.

We choose $u_{n}\in{\mathbb{N}}$ such that the following conditions are
satisfied.

  1. (a)

$u_{n}\geq\text{deg}(P_{n-1})+1$;

  2. (b)

$\sum_{j\geq u_{n}}\sum_{k\geq j}2k^{2}r_{n}^{k}\leq\varepsilon_{n}$;

  3. (c)

$(r_{n}/R_{n+1})^{u_{n}}\leq 1-r_{n}/R_{n+1}$.

Note that (b) is possible since $\sum_{j\geq n}\sum_{k\geq j}k^{2}r^{k}\to 0$
as $n\to\infty$ for any $r\in(0,1)$ by Lemma 4.6. Then we apply Mergelyan’s
theorem to define $P_{n}=\sum_{k\geq u_{n}}^{\text{deg}(P_{n})}a_{k,n}z^{k}$
as a polynomial so that

  1. (d)

$\sup_{|z|\leq R_{n}}|P_{n}(z)|\leq\varepsilon_{n}$;

  2. (e)

$\sup_{z\in
K_{\beta(n)}}|P_{n}(r_{n}z)-\left(\varphi_{\alpha(n)}(z)-\sum_{0\leq j\leq
n-1}(P_{j}+Q_{j})(r_{n}z)\right)|\leq\varepsilon_{n}$

It remains to define $Q_{n}$, which is the most important step of the
construction. To proceed, we will build by a finite induction
$\text{deg}(P_{n})-u_{n}+2$ complex numbers $b_{u_{n}-1,n}$,
$b_{u_{n},n},\ldots,b_{\text{deg}(P_{n}),n}$. First we set $b_{u_{n}-1,n}=0$.
Let us then assume that $b_{u_{n}-1,n}$, $b_{u_{n},n},\ldots,b_{l-1,n}$ have
been chosen for some $l\in\\{u_{n},\ldots,\text{deg}(P_{n})\\}$. To define
$b_{l,n}$ we apply Lemma 4.14 with
$w_{m}=\sum_{j=0}^{n-1}(P_{j}+Q_{j})(\zeta_{m})+\sum_{k=u_{n}}^{l-1}(a_{k,n}+b_{k,n})\zeta_{i}^{k}+a_{l,n}\zeta_{m}^{l},z_{m}=\zeta_{m}^{l}$
for $m=1,\dots,l$ and $R=l$. So, there exist $b_{l,n}\in\mathbb{C}$ with
$|b_{l,n}|\leq 2l^{2}$ such that
$|\sum_{j=0}^{n-1}(P_{j}+Q_{j})(\zeta_{m})+\sum_{k=u_{n}}^{l-1}(a_{k,n}+b_{k,n}){\zeta_{m}}^{k}+a_{l,n}{\zeta_{m}}^{l}+b_{l,n}{\zeta_{m}}^{l}|\geq
l$ for $m=1,\dots,l$. We set

$Q_{n}=\sum_{k=u_{n}-1}^{\text{deg}(P_{n})}b_{k,n}z^{k}=\sum_{k=u_{n}}^{\text{deg}(P_{n})}b_{k,n}z^{k}$

to conclude the induction.

Now we have to check that the power series $f=\sum_{n}(P_{n}+Q_{n})$ satisfies
the desired properties. First note by (d) that the series $\sum_{n}P_{n}$
defines a function in $H({\mathbb{D}})$. Moreover, the construction ensures
that for every $n$ and every $k$, $|b_{k,n}|\leq 2k^{2}$, so that
$\sum_{n}Q_{n}$ also belongs to $H({\mathbb{D}})$. Thus $f$ is well-defined in
$H({\mathbb{D}})$.

Now, we fix $m\in{\mathbb{N}}$. We want to show that
$S_{n}(f)(\zeta_{m})\to\infty$ as $n\to\infty$. Let

$I=\cup_{n\geq 1}\\{u_{n},u_{n}+1,\ldots,\text{deg}(P_{n})\\}.$

We note that for every $n\geq m,n\in I$, we have that
$|S_{n}(f)(\zeta_{m})|\geq n$ and so $S_{n}(f)(\zeta_{m})\to\infty$ as
$n\to\infty,n\in I$. Also, for every $n\geq 1$ and every
$N\in\\{\text{deg}(P_{n}),\ldots,u_{n+1}\\}$, we have
$S_{N}(f)(z)=S_{\text{deg}(P_{n})}(f)(z)$ for any $z\in{\mathbb{T}}$.
Therefore $S_{N}(f)(\zeta_{m})\to\infty$ as $N\to\infty$.

To finish the proof, it remains to check that $f$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Let us fix $n,m\in{\mathbb{N}}$ and
let $(n_{l})_{l}\subset{\mathbb{N}}$ be such that for any $l\in{\mathbb{N}}$,
$\alpha(n_{l})=n$ and $\beta(n_{l})=m$. By (e) we have, for any $z\in K_{m}$,

(7)
$\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\leq\varepsilon_{n_{l}}+\left|Q_{n_{l}}(r_{n_{l}}z)\right|+\left|\sum_{j\geq
n_{l}+1}\left(P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right)\right|.$

By construction of the $Q_{n}$ (and the $b_{k,n}$) and the choice of the
$u_{n}$ (see (b)), we have for any $z\in{\mathbb{T}}$,

$\left|Q_{n_{l}}(r_{n_{l}}z)\right|\leq\sum_{k=u_{n_{l}}}^{\text{deg}(P_{n_{l}})}2k^{2}r_{n_{l}}^{k}\leq\varepsilon_{n_{l}}.$

To deal with the third term of the right hand-side of (7), we derive first
from (d) and Cauchy’s inequalities that for any $n\geq 1$ and any
$k\in\\{u_{n},\ldots,\text{deg}(P_{n})\\}$,
$|a_{k,n}|\leq\varepsilon_{n}R_{n}^{-k}$. Then, together with the definition
of the $b_{k,n}$, conditions (b) and (c) we obtain, for any $z\in K_{m}$,

$\displaystyle\left|\sum_{j\geq
n_{l}+1}\left(P_{j}(r_{n_{l}}z)+Q_{j}(r_{n_{l}}z)\right)\right|$
$\displaystyle\leq$ $\displaystyle\sum_{j\geq n_{l}+1}\sum_{k\geq
u_{j}}\varepsilon_{j}\left(\frac{r_{n_{l}}}{R_{j}}\right)^{k}+\sum_{j\geq
n_{l}+1}\sum_{k\geq u_{j}}2k^{2}r_{n_{l}}^{k}$ $\displaystyle\leq$
$\displaystyle\sum_{j\geq
n_{l}+1}\varepsilon_{j}\frac{\left(r_{n_{l}}/R_{n_{l}+1}\right)^{u_{n_{l}}}}{1-r_{n_{l}}/R_{n_{l}+1}}+\sum_{j\geq
u_{n_{l}}}\sum_{k\geq j}2k^{2}r_{n_{l}}^{k}$ $\displaystyle\leq$
$\displaystyle\sum_{j\geq n_{l}+1}\varepsilon_{j}+\varepsilon_{n_{l}}.$

Altogether we get

$\sup_{z\in K_{m}}\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\rightarrow
0\quad\text{as }l\to\infty,$

and thus $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

######  Remark 4.15.

Note that the sets of functions in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ with
the properties described in the results of this section are dense in
$H({\mathbb{D}})$ since they are invariant under addition with a polynomial.
However, they are of first category in $H({\mathbb{D}})$ because they all
consist of functions that are not universal Taylor series. (Recall that a
subset $F$ of a topological space $X$ is said to be of first category in $X$
if $F$ can be written as a countable union of subsets which are nowhere dense
in $X$.)

The following questions remain open:

######  Questions 4.16.

(i) Is it true that for any Abel universal function $f$ there exists
$z\in{\mathbb{T}}$ such that $\\{S_{n}(f)(z):\,n\in{\mathbb{N}}\\}$ is dense
in ${\mathbb{C}}$?

(ii) In the opposite direction, does there exist an Abel universal function
$f$ such that $S_{n}(f)\to\infty$ everywhere on ${\mathbb{T}}$?

(iii) If the answer in (ii) is negative, in Corollary 4.13, can we replace
countable by first category?

##  5\. Further developments

In this final section, we will focus on extensions of the results obtained in
the previous parts to other types of universal functions, that are naturally
related to Abel universal functions. We mentioned in the introduction three
such classes of functions in $H({\mathbb{D}})$ with a wild boundary behaviour.
Let us recall the definitions. For $f\in H({\mathbb{D}})$ and
$\gamma:[0,1)\to{\mathbb{D}}$ a path in ${\mathbb{D}}$ with
$\gamma(r)\to\zeta$ as $r\to 1$ for some $\zeta\in{\mathbb{T}}$, the cluster
set $C_{\gamma}(f)$ of $f$ along $\gamma$ is defined by

$C_{\gamma}(f)=\left\\{w\in\mathbb{C}\cup\\{\infty\\}:\,w=\lim_{n\to\infty}f\left(\gamma(r_{n})\right)\text{for
some }(r_{n})_{n}\text{ in }[0,1)\text{ with }r_{n}\to 1\right\\}.$

We will also denote by $r_{\zeta}:=\\{r\zeta:\,r\in[0,1)\\}$ the radius
through $\zeta$ and by $C_{r_{\zeta}}(f)$ the radial cluster set through the
radius $r_{\zeta}$. Finally, we say that a cluster set is maximal if it is
equal to $\mathbb{C}\cup\\{\infty\\}$.

######  Definition 5.1.

The sets ${\mathcal{U}}_{R}({\mathbb{D}})$,
${\mathcal{U}}_{\Gamma}({\mathbb{D}})$ and ${\mathcal{U}}_{ae}({\mathbb{D}})$
consist of all functions $f$ in $H({\mathbb{D}})$ that satisfy the following
properties respectively:

  1. (a)

$C_{r_{\zeta}}(f)$ is maximal for any $\zeta\in{\mathbb{T}}$;

  2. (b)

$C_{\gamma}(f)$ is maximal for any path $\gamma$ in ${\mathbb{D}}$ with an
endpoint in ${\mathbb{T}}$;

  3. (c)

for every $m$-measurable function $\varphi$ on ${\mathbb{T}}$, there exists a
sequence $(r_{n})_{n}$ in $[0,1)$ such that
$f_{r_{n}}(\zeta)\to\varphi(\zeta)$ for $m$-a.e. $\zeta\in{\mathbb{T}}$.

We recall that ${\mathcal{U}}_{R}({\mathbb{D}})$,
${\mathcal{U}}_{\Gamma}({\mathbb{D}})$ and ${\mathcal{U}}_{ae}({\mathbb{D}})$
were proved to be residual in [32], [8] and [4] respectively (see also [10]
for the establishment of the existence of functions in
${\mathcal{U}}_{\Gamma}({\mathbb{D}})$). It is easy to see that we have the
following inclusions:

${\mathcal{U}}_{A}({\mathbb{D}})\subset{\mathcal{U}}_{\Gamma}({\mathbb{D}})\subset{\mathcal{U}}_{R}({\mathbb{D}})\quad\text{and}\quad{\mathcal{U}}_{A}({\mathbb{D}})\subset{\mathcal{U}}_{ae}({\mathbb{D}}).$

It was proved in [16] that the inclusion
${\mathcal{U}}_{A}({\mathbb{D}})\subset{\mathcal{U}}_{\Gamma}({\mathbb{D}})$
is strict.

It is natural to wonder whether all functions in these three classes satisfy
the same wild boundary properties with Abel universal functions. On the one
hand, it is clear that the set ${\mathcal{U}}_{\Gamma}({\mathbb{D}})$ and the
MacLane class do not intersect. Thus functions in
${\mathcal{U}}_{A}({\mathbb{D}})$ must grow fast near ${\mathbb{T}}$ (see
Theorem 3.2). On the other hand, Bayart [4] in fact proved that given any
growth function $w:[0,1)\to[1,\infty)$ with $w(r)\to\infty$ as $r\to 1$, there
exists $f\in{\mathcal{U}}_{ae}({\mathbb{D}})$ such that $M(r,f)\leq w(r)$. We
also point out that, so far, it is not clear whether
${\mathcal{U}}_{R}({\mathbb{D}})$ may intersect the MacLane class or not. Thus
we cannot use Theorem 3.2 to make conclusions about the minimal growth of
$M(r,f)$ for functions $f$ in ${\mathcal{U}}_{R}({\mathbb{D}})$. Moreover, one
can easily check that the functions in ${\mathcal{U}}_{R}({\mathbb{D}})$
satisfy the assumptions of Theorem 3.5 (a) and (b) and that the functions in
${\mathcal{U}}_{ae}({\mathbb{D}})$ satisfy the assumptions of Theorem 3.5 (a),
but it is not clear if they satisfy the assumptions of (b). Thus, we don’t
have any information about a Picard-type property that functions in
${\mathcal{U}}_{ae}({\mathbb{D}})$ may satisfy. However we can derive the
following result, using tools from potential theory, as in [22, Corollary 6].

######  Theorem 5.2.

Let $f\in H(\mathbb{D})$ be such that $\\{f(rw):0\leq r<1\\}$ is unbounded for
a.e. $w\in{\mathbb{T}}$. Then for all $\zeta\in{\mathbb{T}}$ and $r>0$, the
set $f(D_{\zeta,r})$ has polar complement, where
$D_{\zeta,r}=\\{z\in\mathbb{D}:|z-\zeta|<r\\}$. In particular, this holds if
$f\in{\mathcal{U}}_{ae}(\mathbb{D})$.

###### Proof.

For the sake of contradiction assume that there is $\zeta\in{\mathbb{T}}$ and
$r>0$ such that $f(D_{\zeta,r})$ has non-polar complement. Then, by Myrberg’s
Theorem (see [2, Theorem 5.3.8]), there is a positive harmonic function $h$ on
$f(D_{\zeta,r})$ such that $\log|z|\leq h(z)$ there. Thus, $\log|f|$ is
majorized by the positive harmonic function $h\circ f$ on $D_{\zeta,r}$, and
so (by Fatou’s Theorem) it is nontangentially bounded above at a.e. point in
$\partial D_{\zeta,r}\cap{\mathbb{T}}$, which leads to a contradiction. ∎

All the previous results can be summarized in the following table, where we
use the notation $I(f):=\int_{0}^{1}\log^{+}\log^{+}M(r,f)dr$:

| $\mathcal{U}_{A}(\mathbb{D})$ | $\mathcal{U}_{\Gamma}(\mathbb{D})$ | $\mathcal{U}_{R}(\mathbb{D})$ | $\mathcal{U}_{ae}(\mathbb{D})$
---|---|---|---|---
Growth | $I(f)=\infty$ | $I(f)=\infty$ | ? | arbitrarily slow
Asymp. values | $\infty$ | $\infty$ | $\infty$ | $\infty$
Normality | At no $\zeta\in\mathbb{T}$ | At no $\zeta\in\mathbb{T}$ | At no $\zeta\in\mathbb{T}$ | ?
Picard points | Every $\zeta\in\mathbb{T}$ | Every $\zeta\in\mathbb{T}$ | Every $\zeta\in\mathbb{T}$ |  ? (Theorem 5.2)

######  Question 5.3.

Is it possible to complete the above table?

Another interesting generalisation of $\mathcal{U}_{A}({\mathbb{D}},\rho)$ is
given by restricting the universal approximation to a fixed compact subset of
${\mathbb{T}}$. More precisely:

######  Definition 5.4.

Let $\rho=(r_{n})_{n}$ be a sequence in $[0,1)$ such that $r_{n}\to 1$, and
let $K$ be a compact set in ${\mathbb{T}}$, different from ${\mathbb{T}}$. We
denote by $\mathcal{U}_{A}^{K}(\mathbb{D},\rho)$ the set of all functions
$f\in H({\mathbb{D}})$ for which the family of dilates
$\\{f_{r_{n}}:\,n\in\mathbb{N}\\}$ is dense in $C(K)$.

When $K$ is a non-trivial compact arc in ${\mathbb{T}}$ (different from
${\mathbb{T}}$), the functions in $\mathcal{U}_{A}^{K}(\mathbb{D},\rho)$ have
similar behaviour with the functions in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$
near points of $K$, even if the two classes are different (obviously,
${\mathcal{U}}_{A}({\mathbb{D}},\rho)\subset\mathcal{U}_{A}^{K}(\mathbb{D},\rho)$).
In particular, the results of this paper that hold for classical Abel
universal functions carry over to the elements of the class
${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)$, up to superficial modifications
depending on $K$. The most interesting case concerns compact sets $K$ that
allow functions in ${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)$ to belong to
classical function subspaces of $H({\mathbb{D}})$. This case was examined in
[34], where it was proved that if $m(K)=0$, then
${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)\cap H^{p}$ is a dense $G_{\delta}$
subset of the Hardy space $H^{p}$ of the unit disc, for $1\leq p<\infty$.
Similar results were obtained for the Bergman and the Dirichlet spaces.

It would be interesting to examine which properties such Abel universal
functions in $H^{p}$ satisfy, related to their boundary behaviour and the
behaviour of their Taylor expansion. For example, since functions in the Hardy
space belong to the MacLane class, Corollary 3.6 (a) fails for any function in
${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)\cap H^{p}$ (for $m(K)=0$).
Nevertheless, one may wonder whether functions of this class still have a
typical minimal growth. Moreover, it is known that the set of all functions in
$H^{p}$ with the property that every $\zeta\in{\mathbb{T}}$ is a Picard point
is residual [11]. Thus, it natural to ask whether every $\zeta\in K$ is a
Picard point for every function in ${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)\cap
H^{p}$. Furthermore, regarding the results of Section 4, the following
question arises: _given $K\subset{\mathbb{T}}$ with $m(K)=0$, do the Taylor
partial sums of the functions of ${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)\cap
H^{p}$ behave chaotically on $K$?_ We know that this is the case for most
functions in ${\mathcal{U}}_{A}^{K}(\mathbb{D},\rho)\cap H^{p}$ (see [5]). We
note that in the opposite direction, it is known that if $K$ is finite, there
exists a function $f$ in the disc algebra $A(\mathbb{D})$ whose Taylor partial
sums are universal on $K$ (see [43] for a precise statement). Similarly, under
different sufficient conditions on $K$, it is also known that
${\mathcal{U}}_{A}^{K}({\mathbb{D}},\rho)\bigcap A^{p}\neq\emptyset$ and
${\mathcal{U}}_{A}^{K}({\mathbb{D}},\rho)\bigcap{\mathcal{D}}\neq\emptyset$,
where $A^{p}$ is the Bergman space for $1\leq p<\infty$ and ${\mathcal{D}}$ is
the Dirichlet space (see [34]). The analogues of the above questions can be
asked for these spaces.

To finish let us focus on the Bloch space $\mathcal{B}$, another classical
Banach space of analytic functions, of particular interest for the study of
univalent functions. We recall that a function $f\in H({\mathbb{D}})$ belongs
to the Bloch space $\mathcal{B}$ if

$\|f\|_{\mathcal{B}}:=\sup_{z\in\mathbb{{\mathbb{D}}}}(1-|z|^{2})|f^{\prime}(z)|<\infty.$

We introduce the following definition:

######  Definition 5.5.

Let $K$ be a compact subset of ${\mathbb{T}}$. We denote by
${\mathcal{U}}_{R}^{K}({\mathbb{D}})$ the set of all functions $f\in
H({\mathbb{D}})$ for which the set $C_{r_{\zeta}}(f)$ is maximal for any
$\zeta\in K$.

Observe that ${\mathcal{U}}_{A}^{K}({\mathbb{D}}\
\rho)\subset{\mathcal{U}}_{R}^{K}({\mathbb{D}})$ for any $\rho$ and
$K\subset\mathbb{T}$. The first part of the next theorem shows the contrast
between the Bloch space and the other classical spaces mentioned above.

######  Theorem 5.6.

We have the following:

  1. (a)

${\mathcal{U}}_{R}({\mathbb{D}})\bigcap\mathcal{B}=\emptyset$.

  2. (b)

${\mathcal{U}}_{R}^{\mathbb{T}\setminus
E}({\mathbb{D}})\bigcap\mathcal{B}\neq\emptyset$, for some $E$ in $\mathbb{T}$
with $m(E)=0$.

###### Proof.

(a) is a consequence of Theorem 3.5 (b) and the fact that each function in
$\mathcal{B}$ is normal (see p. 71 in [45]).

(b) It is known that there is a function $f$ in $\mathcal{B}$ that has radial
limits at no point of $\mathbb{T}$. For example, one can consider the function
$f(z)=\sum_{n=1}^{\infty}z^{2^{n}}$ which is in $\mathcal{B}$ as a Hadamard
lacunary series with bounded coefficients. By the Tauberian theorem of Hardy
and Littlewood (which states that if a Hadamard lacunary power series has a
radial limit at a point then the power series converges at that point), $f$
has no radial limits. We will now make use of a stronger version of Plessner’s
Theorem for functions in the Bloch space. More precisely, [45, Corollary 6.15]
tells us that for any function $g\in\mathcal{B}$ we have that for $m$-a.e.
$\zeta\in\mathbb{T}$ either $g$ has a finite nontangential limit at $\zeta$ or
the cluster set of $g$ along $r_{\zeta}$ is maximal. Thus, applying this to
$f$, we get that for some set $E$ in $\mathbb{T}$ with zero arclength measure
we have that $f\in{\mathcal{U}}_{R}^{\mathbb{T}\setminus
E}({\mathbb{D}})\bigcap\mathcal{B}$. ∎

This result suggests the following question:

######  Question 5.7.

In Theorem 5.6 (b), can we replace ${\mathcal{U}}_{R}^{\mathbb{T}\setminus
E}(\mathbb{D})$ by the smaller class ${\mathcal{U}}_{A}^{\mathbb{T}\setminus
E}(\mathbb{D},\rho)$?

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