# Iterates of Blaschke products and Peano curves

Juan Jesús Donaire and Artur Nicolau
Departament de Matemàtiques, Universitat Autònoma de Barcelona and Centre de
Recerca Matemàtica, 08193 Barcelona<EMAIL_ADDRESS><EMAIL_ADDRESS>The
authors are supported in part by the Generalitat de Catalunya (grant 2017 SGR
395), the Spanish Ministerio de Ciencia e Innovación (project MTM2017-85666-P)
and the María de Maeztu Award.

###### Abstract

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation and
let $f^{n}$ be its $n$-th iterate. Given a sequence $\\{a_{n}\\}$ of complex
numbers consider $F=\sum a_{n}f^{n}$. If $\\{a_{n}\\}$ tends to $0$ but
$\sum|a_{n}|=\infty$, we prove that for any complex number $w$ there exists a
point $\xi$ in the unit circle such that $\sum a_{n}f^{n}(\xi)$ converges and
its sum is $w$. If $\sum|a_{n}|<\infty$ and the convergence is slow enough in
a certain precise sense, then the image of the unit circle by $F$ has a non
empty interior. The proofs are based on inductive constructions which use the
beautiful interplay between the dynamics of $f$ as a selfmapping of the unit
circle and those as a selfmapping of the unit disc.

##  1 Introduction

A lacunary power series is a power series of the form

(1) $F(z)=\sum_{n=1}^{\infty}a_{n}z^{k_{n}},$

where $\\{a_{n}\\}$ is a sequence of complex numbers and $\\{k_{n}\\}$ is a
sequence of positive integers satisfying $\inf k_{n+1}/k_{n}>1$. The behavior
of lacunary series has been extensively studied and it has been shown that in
many senses, they behave as sums of independent random variables. If the
coefficients $\\{a_{n}\\}$ satisfy $\sum|a_{n}|=\infty$ but $a_{n}\to 0$ as
$n\to\infty$, a theorem of Paley, proved by Weiss in [W], says that for any
$w\in\mathbb{C}$ there exists a point $\xi$ in the unit circle
$\partial\mathbb{D}$ such that $\sum a_{n}\xi^{k_{n}}$ converges and its sum
is $w$. Salem and Zygmund proved that boundary values of certain lacunary
series are Peano curves ([SZ]). Their result was refined by Kahane, Weiss and
Weiss who showed that if $\sum|a_{n}|<\infty$ but the convergence is slow
enough (in a certain precise sense), then
$F\colon\partial\mathbb{D}\to\mathbb{C}$ as defined in (1) is a Peano curve,
that is, $F(\partial\mathbb{D})$ contains a (non-degenerate) disc. See [KWW].
More recent related results have been proved by Barański ([Ba]), Belov ([Be]),
Murai ([Mu1], [Mu2], [Mu3]) and Younsi ([Y]).

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation and
let $f^{n}$ denote its $n$-th iterate. The main purpose of this paper is to
present analogous results for series of the form

(2) $F(z)=\sum_{n=1}^{\infty}a_{n}f^{n}(z).$

It is worth mentioning that several recent results show that linear
combinations of iterates, as defined in (2), behave as lacunary series. See
[NS] and [N]. Our first result is a version of Paley’s Theorem in this
context.

######  Theorem 1.1.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Let $\\{a_{n}\\}$ be a sequence of complex numbers tending to $0$ such that
$\sum|a_{n}|=\infty$. Then for any $w\in\mathbb{C}$ there exists
$\xi\in\partial\mathbb{D}$ such that $\sum a_{n}f^{n}(\xi)$ converges and
$\displaystyle{\sum_{n=1}^{\infty}}a_{n}f^{n}(\xi)=w$.

Let $m$ denote Lebesgue measure on the unit circle normalized so that
$m(\partial\mathbb{D})=1$. It has been proved in [N] that if
$\sum|a_{n}|^{2}<\infty$, then the series (2) converges at almost every point
of the unit circle. Conversely, if $\sum|a_{n}|^{2}=\infty$, then the series
(2) diverges at almost every point of the unit circle. Hence if the
coefficients $\\{a_{n}\\}$ tend to $0$ but $\sum|a_{n}|^{2}=\infty$, Theorem
1.1 provides a set $E\subset\partial\mathbb{D}$ with $m(E)=0$ such that for
any $w\in\mathbb{C}$ there exists $\xi\in E$ such that $\sum a_{n}f^{n}(\xi)$
converges and its sum is $w$. Our next result says that under appropriate
conditions on the coefficients $\\{a_{n}\\}$, series of the form (2) lead to
Peano curves.

######  Theorem 1.2.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Let $\\{a_{n}\\}$ be a sequence of complex numbers with $\sum|a_{n}|<\infty$.
Assume

(3) $\lim_{n\to\infty}\frac{|a_{n}|}{\sum\limits_{k>n}|a_{k}|}=0.$

Then
$F=\displaystyle\sum_{n=1}^{\infty}a_{n}f^{n}\colon\partial\mathbb{D}\to\mathbb{C}$
is a Peano curve, that is, $F(\partial\mathbb{D})$ contains a (non-degenerate)
disc.

The proofs of these results are based on delicate inductive constructions
which use some techniques due to Weiss ([W]) but we also need several new
ideas which arise from the beautiful interplay between dynamical properties of
a Blaschke product as a selfmapping of $\partial\mathbb{D}$ and those as a
selfmapping of the unit disc $\mathbb{D}$. It is worth mentioning that no
lacunarity assumption is needed in our results. The following more technical
result plays a central role in our arguments.

######  Theorem 1.3.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Then there exist constants $\varepsilon=\varepsilon(f)>0$ and $0<c=c(f)<1$
such that the following statement holds. Let $M<N$ be positive integers, let
$z\in\mathbb{D}$ with $|f^{M}(z)|<\varepsilon$ and let $\\{a_{n}:M\leq n\leq
N\\}$ be a collection of complex numbers. Then there exists a point
$\xi\in\partial\mathbb{D}$ with $|\xi-z|\leq c^{-1}(1-|z|)$ such that

(4) $\operatorname{Re}\left(\sum^{N}_{n=M}a_{n}f^{n}(\xi)\right)\geq
c\sum^{N}_{n=M}|a_{n}|.$

The paper is organized as follows. Several auxiliary results are collected in
Section 2. These are used in Section 3 where we present the proofs of our main
results. Finally, in Section 4 we prove a version of the classical Abel’s
Theorem in our context, sketch a proof of a generalization of Theorem 1.1 and
conclude mentioning several open problems.

##  2 Auxiliary results

Let $g$ be an analytic mapping from $\mathbb{D}$ into itself with $g(0)=0$
which is not a rotation. The classical Denjoy-Wolff Theorem says that the
iterates $g^{n}$ converge uniformly to $0$ on compacts of $\mathbb{D}$.
Actually Pommerenke proved the following exponential decay: there exist
constants $0<a=a(f)<1$ such that $|g^{n}(z)|\leq a^{n}(1-|z|)^{-13}$ for any
$z\in\mathbb{D}$. See [P].

A finite Blaschke product $f$ is a finite product of automorphisms of
$\mathbb{D}$, that is,

$f(z)=\prod^{N}_{n=1}\frac{z-z_{n}}{1-\bar{z}_{n}z},\quad z\in\mathbb{C},$

where $z_{1},\dotsc,z_{n}\in\mathbb{D}$ are the zeros of $f$. Observe that

(5)
$i\xi\frac{f^{\prime}(\xi)}{f(\xi)}=\sum^{N}_{n=1}P(z_{n},\xi),\quad\xi\in\partial\mathbb{D},$

where $P(z,\xi)=(1-|z|^{2})|\xi-z|^{-2}$ is the Poisson kernel at the point
$z$. Hence

$f(e^{i\theta})\overline{f(1)}=e^{i\psi(\theta)},\quad 0\leq\theta\leq 2\pi,$

where

$\psi(\theta)=\int^{\theta}_{0}\sum^{N}_{n=1}P(z_{n},e^{it})\,dt.$

Note that $\psi$ is a real analytic branch of the argument of
$f(e^{i\theta})\overline{f(1)}$ which is increasing and satisfies

(6)
$f(e^{i\theta})\overline{f(e^{i\varphi})}=\exp\left(i\int^{\theta}_{\varphi}\sum^{N}_{n=1}P(z_{n},e^{it})\,dt\right),\quad
0\leq\varphi\leq\theta\leq 2\pi.$

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Note that (5) gives that
$\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}>1$ and the mapping
$f\colon\partial\mathbb{D}\to\partial\mathbb{D}$ is expanding in the sense
that $m(f(I))>m(I)$ for any arc $I\subset\partial\mathbb{D}$ with $m(I)<1$.
Our first auxiliary result points in this direction.

######  Lemma 2.1.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Consider $K=K(f)=\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}>1$. Let
$N$ be a positive integer and let $I\subset\partial\mathbb{D}$ be an arc such
that $m(f^{N}(I))=\delta<1$. Then

(7) $|f^{k}(\xi)-f^{k}(\xi^{\prime})|\leq 2\pi\delta K^{k-N}$

for any $\xi,\xi^{\prime}\in I$ and any integer $0<k\leq N$.

###### Proof.

Since there exists an increasing continuous branch of
$\operatorname{Arg}(f^{N})$ and $m(f^{N}(I))<1$, the mapping $f^{N}\colon I\to
f^{N}(I)$ is one to one. Then for any integer $0<k\leq N$ we have

$\begin{split}\delta&=\int_{I}|(f^{N})^{\prime}|\,dm=\int_{I}|(f^{N-k})^{\prime}(f^{k})||(f^{k})^{\prime}|\,dm\\\
&\geq K^{N-k}\int_{I}|(f^{k})^{\prime}|\,dm=K^{N-k}m(f^{k}(I)).\end{split}$

We deduce that $m(f^{k}(I))\leq\delta K^{k-N}$. Since
$f^{k}(I)\subset\partial\mathbb{D}$ is an arc, the estimate (7) follows. ∎

For future reference we now state two easy consequences of Lemma 2.1.

######  Corollary 2.2.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Then there exists a constant $c=c(f)>0$ such that if
$I\subset\partial\mathbb{D}$ is an arc, $M<N$ are positive integers with
$m(f^{N}(I))=\delta<1$ and $\\{a_{n}:M\leq n\leq N\\}$ is a collection of
complex numbers, then

$\left|\sum^{N}_{n=M}a_{n}(f^{n}(\xi)-f^{n}(\xi^{\prime}))\right|\leq
c\delta\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2},\quad\xi,\xi^{\prime}\in
I.$

###### Proof.

Consider $K=\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}$. Since
$f(0)=0$ and $f$ is not a rotation, identity (5) gives $K>1$. Lemma 2.1 and
Cauchy–Schwarz inequality give

$\sum^{N}_{n=M}|a_{n}||f^{n}(\xi)-f^{n}(\xi^{\prime})|\leq\frac{2\pi\delta}{(K^{2}-1)^{1/2}}\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2}.$

Taking $c=2\pi(K^{2}-1)^{-1/2}$, the proof is completed. ∎

######  Corollary 2.3.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Fix $0<\delta<1$. Let $\\{a_{n}\\}$ be a sequence of complex numbers tending
to $0$. For $N=1,2,\dotsc$, let $I_{N}\subset\partial\mathbb{D}$ be an arc
such that $m(f^{N}(I_{N}))=\delta$. Then

$\max\left\\{\left|\sum^{N}_{n=1}a_{n}(f^{n}(\xi)-f^{n}(\xi^{\prime}))\right|:\xi,\xi^{\prime}\in
I_{N}\right\\}\underset{N\to\infty}{\longrightarrow}0.$

###### Proof.

Consider $K=\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}$. As before,
identity (5) gives $K>1$. Let $L=L(N)$ be a positive integer to be fixed later
satisfying $1<L<N$. Since $m(f^{N}(I_{N}))=\delta$, Lemma 2.1 gives

$\begin{split}\sum^{N}_{n=1}|a_{n}||f^{n}(\xi)-f^{n}(\xi^{\prime})|&\leq
2\pi\delta\sup_{n}|a_{n}|\sum^{L}_{n=1}K^{n-N}+2\pi\delta\sup_{n>L}|a_{n}|\sum^{N}_{n=L+1}K^{n-N}\\\\[5.0pt]
&\leq
2\pi\delta\sup_{n}|a_{n}|\frac{K^{L-N+1}}{K-1}+2\pi\delta\sup_{n>L}|a_{n}|\frac{K}{K-1}\end{split}$

which tends to $0$ as $N$ tends to $\infty$, if $L$ is chosen such that both
$L$ and $N-L$ tend to $\infty$ as $N$ tends to $\infty$. ∎

Given a point $z\in\mathbb{D}\backslash\\{0\\}$, let $I(z)$ denote the arc of
the unit circle centered at $z/|z|$ with $m(I(z))=1-|z|$. Conversely given an
arc $I\subsetneq\partial\mathbb{D}$ let $z(I)$ be the point in $\mathbb{D}$
satisfying $I(z(I))=I$. Let $\rho(z,w)$ denote the pseudohyperbolic distance
between $z,w\in\mathbb{D}$ given by

$\rho(z,w)=\frac{|z-w|}{|1-\bar{w}z|}.$

######  Lemma 2.4.

  1. (a)

For any $0<\gamma<1$, there exists $\delta=\delta(\gamma)>0$ such that if $f$
is a finite Blaschke product and $z\in\mathbb{D}$ satisfies
$|f(z)|\leq\gamma$, then $m(f(I(z)))\geq\delta$.

  2. (b)

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Then, given $0<\delta<1$, there exists $0<\gamma=\gamma(\delta,f)<1$ such that
if $N$ is a positive integer, $I\subset\partial\mathbb{D}$ is an arc with
$m(f^{N}(I))=\delta$, then $|f^{N}(z(I))|\leq\gamma$.

######  Remark 2.5.

It is worth mentioning that the converse of the estimate in (a) does not hold,
that is, $|f(z)|$ could be arbitrarily close to $1$ and $m(f(I(z)))\geq 1/2$.
For instance, if $f(z)=(z-r)^{n}/(1-rz)^{n}$, where $0<r<1$ and $n>2$, we have
$|f(s)|=|(s-r)/(1-rs)|^{n}$, which could be arbitrarily close to $1$ if
$\rho(s,r)$ is sufficiently close to $1$, while $f(I(s))=\partial\mathbb{D}$
if $s<r$. Part (b) says that the converse of the estimate (a) holds uniformly
for iterates of a finite Blaschke product $f$, if the constant $\gamma$ is
allowed to depend on $f$.

###### Proof.

(a) Let $\\{z_{n}\\}$ be the zeros of $f$. We can assume that the mapping
$f\colon I(z)\to f(I(z))$ is one to one since otherwise using (6) we would
deduce that $f(I(z))=\partial\mathbb{D}$. We can also assume
$\inf\\{\rho(z,z_{n}):n\geq 1\\}\geq 1/2$. The identity (5) gives

$m(f(I(z)))=\int_{I(z)}\sum_{n}P(z_{n},\xi)\,dm(\xi).$

Note that there exists a universal constant $c_{1}>0$ such that $|\xi-
z_{n}|\leq c_{1}|1-z_{n}\bar{z}|$ for any $\xi\in I(z)$ and any $n$. We deduce

(8) $m(f(I(z)))\geq
c_{1}^{-2}\sum_{n}\frac{1-|z_{n}|^{2}}{|1-z_{n}\bar{z}|^{2}}(1-|z|).$

Since $\inf\\{\rho(z,z_{n}):n\geq 1\\}\geq 1/2$, the elementary estimate
$-\log x\leq c_{2}(1-x^{2})$, $1/2\leq x\leq 1$, provides a universal constant
$c_{3}>0$ such that

(9) $\sum_{n}\frac{(1-|z_{n}|^{2})(1-|z|)}{|1-z_{n}\bar{z}|^{2}}\geq
c_{3}\log|f(z)|^{-1}.$

This finishes the proof of (a). We now prove (b). We first show that there
exists a constant $c_{4}=c_{4}(f,\delta)>1$ such that

(10) $|(f^{N})^{\prime}(\xi)|\leq
c_{4}|(f^{N})^{\prime}(\xi^{*})|,\quad\xi,\xi^{*}\in I,\quad N\geq 1.$

Using that $|\mathrm{log}x-\log y|\leq|x-y|$ for any $x,y>1$, we obtain

$\begin{split}\left|\log\frac{|(f^{N})^{\prime}(\xi)|}{|(f^{N})^{\prime}(\xi^{*})|}\right|&=\left|\sum^{N-1}_{k=1}\log\frac{|f^{\prime}(f^{k}(\xi))|}{|f^{\prime}(f^{k}(\xi^{*}))|}\right|\\\\[5.0pt]
&\leq\sum^{N-1}_{k=1}|f^{\prime}(f^{k}(\xi))-f^{\prime}(f^{k}(\xi^{*}))|\leq
c_{5}\sum^{N-1}_{k=1}|f^{k}(\xi)-f^{k}(\xi^{*})|,\end{split}$

where $c_{5}=\max\\{|f^{\prime\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}$.
Since $m(f^{N}(I))=\delta<1$, Lemma 2.1 gives the estimate (10).

Let $\xi(I)$ be the center of $I$. Applying (10), for any measurable subset
$J\subset I$, we have

$m(f^{N}(J))=\int_{J}|(f^{N})^{\prime}|\,dm\geq
c_{4}^{-1}|(f^{N})^{\prime}(\xi(I))|m(J)\geq
c_{4}^{-2}m(f^{N}(I))\frac{m(J)}{m(I)}=c_{4}^{-2}\delta\frac{m(J)}{m(I)}.$

We deduce that there exists a constant $c_{6}=c_{6}(\delta,f)>0$ such that

(11) $\frac{1}{m(I)}\int_{I}|f^{N}-f^{N}(z(I))|^{2}\,dm\geq c_{6}.$

Since there exists a universal constant $c_{7}>0$ such that $P(z(I),\xi)\geq
c_{7}m(I)^{-1}$ for any $\xi\in I$, we obtain

$\begin{split}\frac{1}{m(I)}\int_{I}|f^{N}-f^{N}(z(I))|^{2}\,dm&\leq
c_{7}^{-1}\int_{\partial\mathbb{D}}|f^{N}(\xi)-f^{N}(z(I))|^{2}P(z(I),\xi)\,dm(\xi)\\\
&=c_{7}^{-1}(1-|f^{N}(z(I))|^{2}).\end{split}$

Using (11) we deduce $1-|f^{N}(z(I))|^{2}\geq c_{6}c_{7}$ and the proof is
completed. ∎

Next auxiliary result will be used in Section 4 and it is not needed in the
proofs of our main results.

######  Lemma 2.6.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Fix $0<\delta<(10\max\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\})^{-1}$.
Then there exits a constant $C=C(\delta,f)>0$ such that the following
statement holds. Given $z\in\mathbb{D}$ let $N(z)$ be the smallest positive
integer such that $m(f^{N(z)}(I(z)))\geq\delta$. Then

(12) $\sum^{N(z)}_{k=1}|f^{k}(\xi)-f^{k}(z)|\leq C,\quad\xi\in I(z).$

###### Proof.

Let $K_{1}=\max\\{|f^{\prime}(\xi)|\,:\,\xi\in\partial\mathbb{D}\\}$. Note
that $N(z)\to\infty$ as $|z|\to 1$ and $m(f^{N(z)}(I(z)))\leq\delta K_{1}$.
Part (b) of Lemma 2.4 gives a constant $0<\gamma=\gamma(\delta,f)<1$ such that
$|f^{N(z)}(z)|\leq\gamma$. Note that $f^{N(z)}$ can not have zeros at
pseudohyperbolic distance less than $1/2$ of $z$, because in this case
$m(f^{N(z)}(I(z)))$ would be large. Then, estimates (8) and (9) in the proof
of part (a) of Lemma 2.4 give that $\delta K_{1}\geq m(f^{N(z)}(I(z)))\geq
c_{1}^{-2}c_{3}\log|f^{N(z)}(z)|^{-1}$. Hence

(13) $e^{-\delta K_{1}c_{1}^{2}/c_{3}}\leq|f^{N(z)}(z)|\leq\gamma.$

Pommerenke estimates of the Denjoy–Wolff Theorem (Lemma 2 of [P]) provide
constants $C(\gamma)>0$ and $0<a=a(f)<1$ such that

(14) $|f^{n}(z)|\leq C(\gamma)a^{n-N(z)},\quad n\geq N(z).$

Let $K=\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}$. Since $f(0)=0$
and $f$ is not a rotation, identity (5) gives $K>1$. Lemma 2.1 gives

$|f^{k}(\xi)-f^{k}(\xi^{\prime})|\leq 2\pi\delta
K_{1}K^{k-N(z)},\quad\xi,\xi^{\prime}\in I(z),\quad 0<k\leq N(z).$

Hence $|f^{k}(\xi)-f^{k}(z)|\leq|f^{k}(\xi^{\prime})-f^{k}(z)|+2\pi\delta
K_{1}K^{k-N(z)}$, for any $\xi,\xi^{\prime}\in I(z)$ and $0<k\leq N(z)$.
Integrating over $\xi^{\prime}\in I(z)$ and using that there exists a constant
$c_{4}>0$ such that $P(z,\xi)\geq c_{4}/(1-|z|),\xi\in I(z)$, we deduce

$\begin{split}|f^{k}(\xi)-f^{k}(z)|&\leq(1-|z|)^{-1}\int_{I(z)}|f^{k}(\xi^{\prime})-f^{k}(z)|\,dm(\xi^{\prime})+2\pi\delta
K_{1}K^{k-N(z)}\\\ &\leq
c_{4}^{-1}\left(\int_{\partial\mathbb{D}}|f^{k}(\xi^{\prime})-f^{k}(z)|^{2}P(z,\xi^{\prime})\,dm(\xi^{\prime})\right)^{1/2}+2\pi\delta
K_{1}K^{k-N(z)}\\\ &=c_{4}^{-1}(1-|f^{k}(z)|^{2})^{1/2}+2\pi\delta
K_{1}K^{k-N(z)},\;\xi\in I(z),\;0<k\leq N(z).\end{split}$

Using Lemma 2.1(a) of [N] and (13) one finds constants $c_{5}=c_{5}(\delta)>0$
and $0<b=b(f)<1$ such that $1-|f^{k}(z)|^{2}\leq c_{5}b^{N(z)-k}$, if $0<k\leq
N(z)$. Hence there exists a constant $C=C(\delta,f)>0$ such that

(15) $\sum^{N(z)}_{k=1}|f^{k}(\xi)-f^{k}(z)|\leq C,\quad\xi\in I(z).$

∎

Next auxiliary result is Lemma 3.3 of [N].

######  Lemma 2.7.

Let $f$ be an inner function with $f(0)=0$ which is not a rotation. Then there
exist two constants $0<\varepsilon=\varepsilon(f)<1$ and $0<c=c(f)<1$ such
that the following statement holds. Let $M<N$ be positive integers,
$z\in\mathbb{D}$ satisfying $|f^{M}(z)|<\varepsilon$ and $\\{a_{n}:M\leq n\leq
N\\}$ a collection of complex numbers. Then there exists a set
$E=E(z,\\{a_{n}\\})\subset\partial\mathbb{D}$ with $E\subset c^{-1}I(z)$ such
that

$\operatorname{Re}\left(\sum^{N}_{n=M}a_{n}f^{n}(\xi)\right)\geq
c\left(\sum^{N}_{n=M}|a_{n}|^{2}\right)^{1/2},\quad\xi\in E.$

Our last auxiliary result is elementary and it is stated for future reference.

######  Lemma 2.8.

Let $\\{b_{n}\\}$ be a sequence of positive numbers with $\sum b_{n}<\infty$.
Consider $S_{n}=\sum\limits_{k>n}b_{k}$. Assume

$\lim_{n\to\infty}\frac{b_{n}}{S_{n}}=0.$

Fix $K>1$. Then

$\lim_{N\to\infty}\frac{\sum_{n=1}^{N}b_{n}K^{n-N}}{S_{N}}=0$

###### Proof.

Fix $\varepsilon>0$ such that $K(1+\varepsilon)^{-1}>1$. By assumption there
exists $n_{0}=n_{0}(\varepsilon)>0$ such that
$S_{n}\geq(1+\varepsilon)^{-1}S_{n-1}$ for any $n\geq n_{0}$. Hence
$S_{n}\geq(1+\varepsilon)^{-n+n_{0}}S_{n_{0}}$, $n\geq n_{0}$. We deduce that

(16) $\lim_{n\to\infty}K^{n}S_{n}=\infty.$

Since $b_{n}\leq\varepsilon S_{n}$ for $n\geq n_{0}$ and
$S_{n}\leq(1+\varepsilon)^{N-n}S_{N}$ for $n_{0}\leq n\leq N$, we have

$\sum^{N}_{n=n_{0}}b_{n}K^{n-N}\leq\varepsilon
S_{N}\sum^{N}_{n=n_{0}}\left(\frac{K}{1+\varepsilon}\right)^{n-N}.$

Now the identity (16) finishes the proof. ∎

##  3 Proofs of the main results

This section is devoted to the proofs of Theorems 1.1, 1.2 and 1.3. It is
worth mentioning that under additional assumptions on the function $f$, one
can give a short proof of Theorem 1.3. Actually, let $f$ be a finite Blaschke
product and assume that there exists a constant $\varepsilon>0$ such that

(17) $f(I)=\partial\mathbb{D}$

for any arc $I\subset\partial\mathbb{D}$ with $m(I)\geq 1/2-\varepsilon$. For
instance, according to (6), any function $f$ of the form $f(z)=z^{2}g(z)$,
where $g$ is a non-constant Blaschke product, satisfies (17). Let
$\\{a_{n}\\}$ be a sequence of complex numbers. For any $n$ such that
$a_{n}\neq 0$, consider the closed arc $I_{n}$ centered at
$\overline{a}_{n}/|a_{n}|$ with $m(I_{n})=1/2-\varepsilon$. Note that there
exists a constant $c=c(\varepsilon)>0$ such that

(18) $\operatorname{Re}\left(\sum_{n}a_{n}\xi_{n}\right)\geq
c\sum_{n}|a_{n}|,$

for any choice of points $\xi_{n}\in I_{n}$, $n\geq 1$. Take $J_{1}=I_{1}$. We
now construct inductively a sequence of nested closed arcs $J_{n}$ such that
$f^{n-1}(J_{n})=I_{n}$, $n>1$. Since $f(I_{1})=\partial\mathbb{D}$, there
exists an arc $J_{2}\subset I_{1}$ with $f(J_{2})=I_{2}$. Assume
$J_{2},\dotsc,J_{n-1}$ have been constructed satisfying
$f^{n-2}(J_{n-1})=I_{n-1}$. Then (17) gives
$f^{n-1}(J_{n-1})=f(I_{n-1})=\partial\mathbb{D}$ and we can find an arc
$J_{n}\subset J_{n-1}$ such that $f^{n-1}(J_{n})=I_{n}$. This finishes the
construction of the nested sequence of arcs $\\{J_{n}\\}$. Pick
$\xi\in\partial\mathbb{D}$ such that $f(\xi)\in\bigcap\limits_{n}J_{n}$. Since
$f^{n}(\xi)\in I_{n}$ for any $n\geq 1$, estimate (18) gives

$\operatorname{Re}\left(\sum_{n}a_{n}f^{n}(\xi)\right)\geq c\sum_{n}|a_{n}|.$

This finishes the proof of Theorem 1.3 under the additional assumption (17).
However there are finite Blaschke products $f$ which do not satisfy (17) and
the proof of Theorem 1.3 requires different ideas.

###### Proof of Theorem 1.3.

The proof of Theorem 1.3 is organized in three steps.

_1\. Splitting._ We use an idea of M. Weiss ([W]). Let $T^{\prime}<T$ be two
(large) positive integers to be fixed later such that $T/T^{\prime}$ is also
an integer. Since one can obviously add terms with vanishing coefficients to
the left-hand side term of (4), one can assume that $N-M+1$ is a multiple of
$T$. Split the sum $F=\sum\limits_{M}^{N}a_{n}f^{n}$ into blocks of length
$T$, that is, $F=\sum\limits_{k\geq 0}G_{k}$, where

$G_{k}=\sum_{n=0}^{T-1}a_{M+kT+n}f^{M+kT+n},\quad k=0,1,2,\dotsc$

Next split $G_{2k}$, $k\geq 1$, into successive blocks of length $T^{\prime}$
and pick the subblock such that the sum of the modulus of the coefficients is
the least, that is, pick $\mathcal{S}_{k}$ a subset of $T^{\prime}$
consecutive integers in $[M+2kT,M+(2k+1)T)$ of the form
$\\{M+2kT+jT^{\prime},\ldots,M+2kT+(j+1)T^{\prime}-1\\}$ such that

$\sum_{n\in\mathcal{S}_{k}}|a_{n}|\leq\sum_{\ell\in\mathcal{S}}|a_{\ell}|,$

for any other subset $\mathcal{S}$ of $T^{\prime}$ consecutive integers in
$[M+2kT,M+(2k+1)T)$ of the same form. Since the number of disjoint subblocks
of this type is $T/T^{\prime}$ we deduce

(19)
$\sum_{n\in\mathcal{S}_{k}}|a_{n}|\leq\frac{T^{\prime}}{T}\sum^{T-1}_{n=0}|a_{M+2kT+n}|,\quad
k\geq 1.$

The corresponding block

${S}_{k}=\sum_{n\in\mathcal{S}_{k}}a_{n}f^{n},\quad k\geq 1$

will be called a short block. The long blocks are defined as the blocks
between two consecutive short blocks as well as the first and last block. More
concretely if $\mathcal{S}_{k}=\\{n\in\mathbb{Z}:N_{k}<n<M_{k+1}\\}$, $k\geq
1$, then

$L_{1}=\sum^{N_{1}}_{n=M}a_{n}f^{n},\quad
L_{k}=\sum^{N_{k}}_{n=M_{k}}a_{n}f^{n},\quad k>1,$

and

$S_{k}=\sum^{M_{k+1}-1}_{n=N_{k}+1}a_{n}f^{n},\quad k\geq 1.$

Write $M_{1}=M$ and $\mathcal{L}_{k}=\\{n\in\mathbb{Z}:M_{k}\leq n\leq
N_{k}\\}$, $k\geq 1$. Note that for any $k\geq 1$, the set $\mathcal{S}_{k}$
has $T^{\prime}$ indexes while the number of indices in $\mathcal{L}_{k}$ is
between $T$ and $3T$. Note also that (19) gives

(20)
$\sum_{k}\sum_{n\in\mathcal{L}_{k}}|a_{n}|\geq\left(1-\frac{T^{\prime}}{T}\right)\sum^{N}_{n=M}|a_{n}|.$

_2\. The inductive construction._ Let $0<\varepsilon=\varepsilon(f)<1$ and
$0<c=c(f)<1$ be the constants appearing in Lemma 2.7. We will show that there
exist constants $c_{0}=c_{0}(f)>0$ and $0<\gamma=\gamma(f)<1$ and a sequence
of nested closed arcs $I_{k}\subset c^{-1}I(z)$, such that

(21) $\displaystyle|f^{N_{k}}(z(I_{k}))|$ $\displaystyle\leq\gamma,$ (22)
$\displaystyle\operatorname{Re}\sum^{k}_{j=1}L_{j}(\xi)$
$\displaystyle\geq\frac{c_{0}}{T^{1/2}}\sum^{k}_{j=1}\sum_{n\in\mathcal{L}_{j}}|a_{n}|,\quad\xi\in
I_{k}.$

We argue by induction. Since $|f^{M}(z)|<\varepsilon$, Lemma 2.7 provides a
point $\xi_{1}\in c^{-1}I(z)$ with

(23) $\operatorname{Re}L_{1}(\xi_{1})\geq
c\left(\sum^{N_{1}}_{n=M}|a_{n}|^{2}\right)^{1/2}.$

Note that $|f^{N_{1}}(z)|\leq|f^{M}(z)|<\varepsilon$. By part (a) of Lemma
2.4, there exists a constant $\delta_{0}=\delta_{0}(\varepsilon)>0$ such that
$m(f^{N_{1}}(I(z)))\geq\delta_{0}$. Fix $0<\delta_{1}<\delta_{0}/2$ and pick
an arc $I_{1}$ with $\xi_{1}\in I_{1}\subset c^{-1}I(z)$ with
$m(f^{N_{1}}(I_{1}))=\delta_{1}$. If $\delta_{1}>0$ is chosen sufficiently
small, Corollary 2.2 and (23) give

$\operatorname{Re}L_{1}(\xi)\geq\frac{c}{2}\left(\sum_{n=M}^{N_{1}}|a_{n}|^{2}\right)^{1/2},\quad\xi\in
I_{1}.$

Since $\mathcal{L}_{1}$ has at most $3T$ indexes, Cauchy–Schwarz’s inequality
gives

$\sum_{n\in\mathcal{L}_{1}}|a_{n}|\leq(3T)^{1/2}\left(\sum_{n\in\mathcal{L}_{1}}|a_{n}|^{2}\right)^{1/2}$

and (22) holds for $k=1$ if we pick $0<c_{0}<c/2\sqrt{3}$. Since
$m(f^{N_{1}}(I_{1}))=\delta_{1}$, part (b) of Lemma 2.4 provides a constant
$\gamma=\gamma(\delta_{1},f)<1$ such that the estimate (21) holds for $k=1$.
Assume now that the arcs $I_{k}\subset I_{k-1}\subset\dotsb\subset I_{1}$ have
been constructed so that (21) and (22) hold. Next we will construct $I_{k+1}$.
Fix a constant $0<c_{1}=c_{1}(c)<1$ such that there exists a point
$z_{k}^{*}\in\mathbb{D}$ with $\rho(z_{k}^{*},z(I_{k}))\leq c_{1}$ with
$c^{-1}I(z_{k}^{*})\subset I_{k}$. Since $|f^{N_{k}}(z(I_{k}))|\leq\gamma$,
Schwarz’s Lemma gives a constant $0<\gamma_{1}=\gamma_{1}(\gamma,c_{1})<1$
with $|f^{N_{k}}(z_{k}^{*})|\leq\gamma_{1}$. Recall that
$M_{k+1}=N_{k}+T^{\prime}$. Since $f^{n}\to 0$ uniformly on compacts of
$\mathbb{D}$, we can choose a positive integer $T^{\prime}=T^{\prime}(f)$
sufficiently large but independent of $k$, so that

$|f^{M_{k+1}}(z_{k}^{*})|\leq\varepsilon.$

Apply Lemma 2.7 to find a point $\xi_{k}^{*}\in c^{-1}I(z_{k}^{*})\subset
I_{k}$ such that

(24) $\operatorname{Re}(L_{k+1}(\xi_{k}^{*}))\geq
c\left(\sum_{n\in\mathcal{L}_{k+1}}|a_{n}|^{2}\right)^{1/2}.$

Note that since $f(0)=0$, Schwarz’s lemma gives
$|f^{N_{k+1}}(z_{k}^{*})|\leq|f^{M_{k+1}}(z_{k}^{*})|\leq\varepsilon$. By part
(a) of Lemma 2.4, there exists a constant
$\delta_{0}=\delta_{0}(\varepsilon)>0$ such that
$m(f^{N_{k+1}}(I(z_{k}^{*})))\geq\delta_{0}$. Fix $0<\delta_{1}<\delta_{0}/2$
and pick an arc $I_{k+1}$ with $\xi_{k}^{*}\in I_{k+1}\subset I_{k}$ such that
$m(f^{N_{k+1}}(I_{k+1}))=\delta_{1}$. If $\delta_{1}>0$ is chosen sufficiently
small, Corollary 2.2 and estimate (24) give

$\operatorname{Re}(L_{k+1}(\xi))\geq\frac{c}{2}\left(\sum_{n\in\mathcal{L}_{k+1}}|a_{n}|^{2}\right)^{1/2},\quad\xi\in
I_{k+1}.$

Since $\mathcal{L}_{k+1}$ has at most $3T$ indexes, Cauchy–Schwarz’s
inequality gives

$\sum_{n\in\mathcal{L}_{k+1}}|a_{n}|\leq(3T)^{1/2}\left(\sum_{n\in\mathcal{L}_{k+1}}|a_{n}|^{2}\right)^{1/2}$

and we deduce

$\operatorname{Re}L_{k+1}(\xi)\geq\frac{c}{2}\frac{1}{(3T)^{1/2}}\sum_{n\in\mathcal{L}_{k+1}}|a_{n}|,\quad\xi\in
I_{k+1}.$

If we pick $0<c_{0}<c/2\sqrt{3}$, the inductive assumption gives

$\operatorname{Re}\sum^{k+1}_{j=1}L_{j}(\xi)\geq\frac{c_{0}}{T^{1/2}}\sum^{k+1}_{j=1}\sum_{n\in\mathcal{L}_{j}}|a_{n}|,\quad\xi\in
I_{k+1}$

which is (22) for the index $k+1$. Since $m(f^{N_{k+1}}(I_{k+1}))=\delta_{1}$,
the estimate (21) for the index $k+1$, follows from part (b) of Lemma 2.4.
This finishes the proof of the existence of the nested sequence of arcs
$I_{k}$ satisfying (21) and (22).

_3\. The final argument._ Pick $\xi\in\bigcap\limits_{k}I_{k}$. The estimate
(22) gives

$\operatorname{Re}\left(\sum^{k}_{j=1}L_{j}(\xi)+\sum^{k-1}_{j=1}S_{j}(\xi)\right)\geq\frac{c_{0}}{T^{1/2}}\sum^{k}_{j=1}\sum_{n\in\mathcal{L}_{j}}|a_{n}|-\sum^{k-1}_{j=1}\sum_{n\in\mathcal{S}_{j}}|a_{n}|,\quad
k>1.$

Since $\sum_{n=M}^{N}a_{n}f^{n}$ is a sum of long and short blocks, the
estimates (19) and (20) give

$\operatorname{Re}\left(\sum_{n=M}^{N}a_{n}f^{n}(\xi)\right)\geq\frac{c_{0}}{T^{1/2}}\left(1-\frac{T^{\prime}}{T}\right)\sum_{n=M}^{N}|a_{n}|-\frac{T^{\prime}}{T}\sum_{n=M}^{N}|a_{n}|.$

Choosing $T$ such that $T^{\prime}/T^{1/2}<c_{0}/4$, we deduce

$\operatorname{Re}\left(\sum_{n=M}^{N}a_{n}f^{n}(\xi)\right)\geq\frac{c_{0}}{2T^{1/2}}\sum_{n=M}^{N}|a_{n}|.$

This finishes the proof. ∎

The proof of Theorem 1.1 uses the following easy consequence of Theorem 1.3.

######  Corollary 3.1.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Then there exists a constant $0<\eta=\eta(f)<1$ such that the following
statement holds. Let $M<N$ be positive integers, let $z\in\mathbb{D}$ with
$|f^{M}(z)|\leq\eta$, let $\\{a_{n}:M\leq n\leq N\\}$ be a collection of
complex numbers and let $w\in\mathbb{C}$ with

(25) $\sum^{N}_{n=M}|a_{n}|\leq\eta|w|.$

Then there exists a point $\xi\in\eta^{-1}I(z)$ such that

$\left|w-\sum^{N}_{n=M}a_{n}f^{n}(\xi)\right|\leq|w|-\eta\sum^{N}_{n=M}|a_{n}|.$

###### Proof.

We can assume $w=1$. Let $0<\varepsilon=\varepsilon(f)<1$ and $0<c=c(f)<1$ be
the constants appearing in Theorem 1.3. Pick $0<\eta<\min\\{\varepsilon,c\\}$.
Applying Theorem 1.3 one finds a point $\xi\in c^{-1}I(z)$ such that

(26) $\operatorname{Re}\left(\sum^{N}_{n=M}a_{n}f^{n}(\xi)\right)\geq
c\sum^{N}_{n=M}|a_{n}|.$

Write $A=\sum\limits_{n=M}^{N}|a_{n}|$. Since $A\leq\eta<1$, we have

$\left|1-\sum^{N}_{n=M}a_{n}f^{n}(\xi)\right|^{2}\leq(1-cA)^{2}+A^{2}-(cA)^{2}=1-A(2c-A).$

See Figure 1. Using the elementary estimate $\sqrt{1-x}\leq 1-x/2$, $0\leq
x<1$, we deduce that
$\bigl{|}1-\sum\limits_{n=M}^{N}a_{n}f^{n}(\xi)\bigr{|}\leq 1-A(c-A/2)\leq
1-\eta A$ if $0<\eta<2c/3$.

Figure 1: $\sum a_{n}f^{n}(\xi)$ lies in the shadowed region. ∎

We are now ready to prove Theorem 1.1. We will again combine some ideas of
Weiss ([W]) with the interplay between dynamical properties of $f$ as a
selfmapping of $\mathbb{D}$ with those as a selfmapping of
$\partial\mathbb{D}$.

###### Proof of Theorem 1.1.

Fix $w\in\mathbb{C}$. We first divide the sum $\sum a_{n}f^{n}$ into blocks

$P_{j}=\sum_{n\in\mathcal{P}_{j}}a_{n}f^{n},\quad
P_{j}^{*}=\sum_{n\in\mathcal{P}_{j}^{*}}a_{n}f^{n},$

where $\mathcal{P}_{j}=\\{n\in\mathbb{Z}:M_{j}\leq n\leq N_{j}\\}$, $M_{1}=1$
and $\mathcal{P}_{j}^{*}=\\{n\in\mathbb{Z}:N_{j}<n<M_{j+1}\\}$, $j\geq 1$.
This splitting is similar to the decomposition used in the proof of Theorem
1.3 but this time the number of indices in $\mathcal{P}_{j}$ may grow as
$j\to\infty$ while, as before, the number of indices in $\mathcal{P}_{j}^{*}$
is a large number independent of $j$ to be chosen later. In other words, the
blocks will be chosen in such a way that $M_{j+1}-N_{j}=T=T(f)$ is large but
independent of $j$, while $N_{j}-M_{j}$ may tend to infinity as $j\to\infty$.
Write

$F_{k}=\sum^{k}_{j=1}P_{j}+\sum^{k-1}_{j=1}P_{j}^{*}=\sum^{N_{k}}_{n=1}a_{n}f^{n}.$

The blocks $P_{j}$ and $P_{j}^{*}$ will be constructed inductively, as will a
sequence of positive numbers $\delta_{k}$ tending to $0$, and a nested
sequence of closed arcs $I_{k}\subset\partial\mathbb{D}$ in such a way that
there exists a constant $0<\gamma=\gamma(f)<1$ such that if
$d_{k}=\min\\{|F_{k}(\xi)-w|:\xi\in I_{k}\\}$, the following estimates hold

(27) $\displaystyle d_{k+1}\leq\gamma d_{k}+\delta_{k},$ (28)
$\displaystyle\sum_{n\in\mathcal{P}_{k}^{*}\cup\mathcal{P}_{k+1}}|a_{n}|\underset{k\to\infty}{\longrightarrow}0,$
(29) $\displaystyle m(f^{N_{k}}(I_{k}))=1/2.$

Once this inductive construction is done, the result follows easily. Actually
pick $\xi\in\bigcap\limits_{j}I_{j}$. Note that (27) gives $d_{k}\to 0$ as
$k\to\infty$. Then, identity (29) and Corollary 2.3 give that $F_{k}(\xi)\to
w$ as $k\to\infty$. Applying (28) we deduce that
$\sum\limits_{n}a_{n}f^{n}(\xi)$ converges and its sum is $w$.

We now explain the construction of the blocks $P_{j}$, $P_{j}^{*}$ and the
arcs $I_{j}$. For $k=1$ one can pick any partial sum $P_{1}$ and any arc
$I_{1}\subset\partial\mathbb{D}$ with $m(f^{N_{1}}(I_{1}))=1/2$. Assume by
induction that the blocks $P_{1},P_{1}^{*},\dotsc,P_{k-1},P_{k-1}^{*},P_{k}$
and the arcs $I_{k}\subset I_{k-1}\subset\dotsb\subset I_{1}$ have been
chosen. We will now define $P_{k}^{*}$, $P_{k+1}$ and the arc $I_{k+1}\subset
I_{k}$ and show that the estimates (27), (28) and (29) hold. Let
$\eta=\eta(f)>0$ be the constant appearing in Corollary 3.1. Fix a constant
$0<c_{1}=c_{1}(\eta)<1$ such that there exists $z_{k}^{*}\in\mathbb{D}$ with
$\rho(z_{k}^{*},z(I_{k}))\leq c_{1}$ such that $\eta^{-1}I(z_{k}^{*})\subset
I_{k}$. Observe that (29) and part (b) of Lemma 2.4 provide a constant
$0<\gamma_{0}=\gamma_{0}(f)<1$ such that
$|f^{N_{k}}(z(I_{k}))|\leq\gamma_{0}$. By Schwarz’s Lemma, there exists a
constant $0<\gamma_{1}=\gamma_{1}(\gamma_{0},\eta)<1$ such that
$|f^{N_{k}}(z_{k}^{*})|\leq\gamma_{1}$. Since $f^{n}\to 0$ uniformly on
compacts of $\mathbb{D}$, if the number of indices $T=T(f)$ of $P_{k}^{*}$ is
chosen sufficiently large (but independent of $k$), we deduce
$|f^{M_{k+1}}(z_{k}^{*})|\leq\eta$, where $M_{k+1}=N_{k}+T$. This defines
$P_{k}^{*}$. This choice of $M_{k+1}$ allows to apply Corollary 3.1 to the
point $z_{k}^{*}$ and any partial sum starting at the index $M_{k+1}$. To
define the next block $P_{k+1}$ we distinguish two cases.

(I) Assume $\max\limits_{j\geq N_{k}}|a_{j}|\leq\eta d_{k}/4$. In this case,
let $N_{k+1}$ be the minimal integer bigger than $M_{k+1}$ such that

$\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|\geq\eta d_{k}/2.$

This defines the block $P_{k+1}$. Since $|a_{N_{k+1}}|\leq\eta d_{k}/4$, the
minimality of $N_{k+1}$ gives

(30) $3\eta d_{k}/4\geq\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|\geq\eta d_{k}/2.$

Let $\xi_{k}\in I_{k}$ such that $|w-F_{k}(\xi_{k})|=d_{k}$. Apply Corollary
3.1 to the point $z_{k}^{*}\in\mathbb{D}$, the value $w-F_{k}(\xi_{k})$ and
the block $P_{k+1}$ to obtain a point
$\xi_{k+1}\in\eta^{-1}I(z_{k}^{*})\subset I_{k}$ such that

$\begin{split}|w-F_{k}(\xi_{k})-P_{k+1}(\xi_{k+1})|&\leq
d_{k}-\eta\sum_{n\in\mathcal{P}_{k+1}}|a_{n}|\\\\[5.0pt] &\leq
d_{k}\left(1-\frac{\eta^{2}}{2}\right).\end{split}$

Hence

$|w-F_{k}(\xi_{k+1})-P_{k+1}(\xi_{k+1})|\leq
d_{k}\left(1-\frac{\eta^{2}}{2}\right)+\alpha_{k},$

where
$\alpha_{k}=\max\\{|F_{k}(\xi)-F_{k}(\xi^{\prime})|\,:\xi,\xi^{\prime}\in
I_{k}\\}$. We deduce

(31) $|w-F_{k+1}(\xi_{k+1})|\leq
d_{k}\left(1-\frac{\eta^{2}}{2}\right)+\alpha_{k}+\sum_{n\in\mathcal{P}_{k}^{*}}|a_{n}|.$

Since $f$ is expanding and $N_{k+1}>N_{k}$, the identity (29) gives
$m(f^{N_{k+1}}(I_{k}))>1/2$. Let $I_{k+1}$ be an arc with $\xi_{k+1}\in
I_{k+1}\subset I_{k}$ and $m(f^{N_{k+1}}(I_{k+1}))=1/2$. We now check (27),
(28) and (29). Note that (29) holds by construction. Define
$\delta_{k}=\alpha_{k}+\sum_{n\in{\cal P}^{*}_{k}}|a_{n}|$. The estimate (31)
gives $d_{k+1}\leq d_{k}(1-\eta^{2}/2)+\delta_{k}$, which is (27) with
$\gamma=1-\eta^{2}/2$. Corollary 2.3 gives that $\alpha_{k}$ tends to zero and
since ${\cal P}^{*}_{k}$ contains a fix number of indices, it follows that
$\delta_{k}$ tends to zero. Finally note that (30) and the fact that $d_{k}$
tends to $0$ as $k$ tends to infinity, give (28).

(II) Assume $\max\limits_{j\geq N_{k}}|a_{j}|>\eta d_{k}/4$. In this case, we
choose $P_{k+1}$ having a single term, that is, $N_{k+1}=M_{k+1}$. Let
$\xi_{k}\in I_{k}$ such that $|w-F_{k}(\xi_{k})|=d_{k}$. As in case (I), let
$I_{k+1}$ be an arc with $\xi_{k}\in I_{k+1}\subseteq I_{k}$ and
$m(f^{N_{k+1}}(I_{k+1}))=1/2$. Hence (29) follows by construction. Since in
this case $\mathcal{P}_{k}^{*}\cup\mathcal{P}_{k+1}$ has $T+1$ elements, (28)
follows from the assumption that $\\{a_{n}\\}$ tends to $0$. Define
$\delta_{k+1}=\left(4{\eta}^{-1}+T+1\right)\max_{j\geq N_{k}}|a_{j}|$. Note
that $\delta_{k}$ tends to $0$ as $k$ tends to infinity. Then

$\begin{split}d_{k+1}\leq|F_{k+1}(\xi_{k})-w|&\leq|F_{k}(\xi_{k})-w|+|P_{k+1}(\xi_{k})|+|P_{k}^{*}(\xi_{k})|\\\\[5.0pt]
&\leq d_{k}+(T+1)\max_{j\geq N_{k}}|a_{j}|\leq\delta_{k+1}.\end{split}$

This gives (27). ∎

The proof of Theorem 1.2 has some similarities with the proof of Theorem 1.1
but requires some new ideas.

###### Proof of Theorem 1.2.

Let $\eta=\eta(f)>0$ be the constant appearing in Corollary 3.1. Fix any
$\xi_{1}\in\partial\mathbb{D}$. Let $N_{1}$ be a positive integer to be fixed
later and consider $L_{1}=\sum\limits^{N_{1}}_{n=1}a_{n}f^{n}$. We will show
that for any $w\in\mathbb{C}$ satisfying

(32) $|w-L_{1}(\xi_{1})|\leq\frac{\eta}{10}\sum^{\infty}_{n=N_{1}+1}|a_{n}|,$

there exists $\xi\in\partial\mathbb{D}$ such that
$\sum\limits^{\infty}_{n=1}a_{n}f^{n}(\xi)$ converges and its sum is $w$.

Take $M_{1}=1$. We first explain the choice of the positive integer $N_{1}$.
Let $0<\gamma_{1}=\gamma_{1}(\eta)<1$ be a number to be fixed later. Since
$f^{n}$ tends to $0$ uniformly on compacts of $\mathbb{D}$, there exists an
integer $T=T(\eta)>0$ such that

(33) $|f^{T}(z)|\leq\eta\quad\text{ if }\quad|z|<\gamma_{1}.$

Also note that the assumption (3) gives

$\lim_{j\to\infty}\frac{\sum_{n=j}^{j+T}|a_{n}|}{\sum_{n=j+T}^{\infty}|a_{n}|}=0.$

Hence there exists an integer $N_{1}>0$ such that

(34)
$\sum_{n=j}^{j+T}|a_{n}|\leq\frac{\eta^{2}}{100}\sum_{n=j+T}^{\infty}|a_{n}|,\quad
j\geq N_{1}$

and

(35) $|a_{j}|\leq\frac{\eta^{2}}{300}\sum_{k>j}|a_{k}|,\quad j\geq N_{1}.$

Fix any $\xi_{1}\in\partial\mathbb{D}$ and let $I_{1}$ be an arc containing
$\xi_{1}$ such that $m(f^{N_{1}}(I_{1}))=1/2$. Fix $w\in\mathbb{C}$ satisfying
(32). Note that

$d_{1}:=\inf\\{|w-L_{1}(\xi)|:\xi\in
I_{1}\\}<\frac{\eta}{10}\sum^{\infty}_{n=N_{1}+1}|a_{n}|.$

By induction we will construct a sequence of nested arcs
$I_{k}\subset\partial\mathbb{D}$ and positive integers $M_{k}<N_{k}$ with
$M_{1}=1$ and $N_{k}<M_{k+1}$, $k\geq 1$, such that the partial sums
$F_{1}=L_{1}$, $\displaystyle
F_{k}=\sum_{j=1}^{k}L_{j}+\sum_{j=1}^{k-1}S_{j}$, $k>1$, where

$L_{j}=\sum^{N_{j}}_{n=M_{j}}a_{n}f^{n},\quad
S_{j}=\sum_{n=N_{j}+1}^{M_{j+1}-1}a_{n}f^{n},\quad j\geq 1,$

satisfy

(36) $\displaystyle m(f^{N_{k}}(I_{k}))=1/2,\quad k\geq 1,$ (37)
$\displaystyle d_{k}:=\inf\left\\{\biggl{|}w-F_{k}(\xi)\biggr{|}:\xi\in
I_{k}\right\\}\leq\frac{\eta}{10}\sum^{\infty}_{n=N_{k}+1}|a_{n}|,\quad k\geq
1,$

Note that this would end the proof. Since $m(f^{N_{k}}(I_{k}))=1/2$, Corollary
2.3 gives

(38) $\max\\{|F_{k}(\xi)-F_{k}(\xi^{\prime})|:\xi,\xi^{\prime}\in
I_{k}\\}\underset{k\to\infty}{\longrightarrow}0.$

Pick $\xi\in\bigcap\limits_{k}I_{k}$. Note that (37) gives that $d_{k}$ tends
to zero. Then (38) gives that $F_{k}(\xi)$ converges to $w$. Since
$\sum\limits_{n}|a_{n}|<\infty$, this would finish the proof.

The construction of the blocks $L_{j}$, $S_{j}$ and the arcs $I_{j}$ is done
by induction and has some similarities to the construction used in the proof
of Theorem 1.1. Again the number of terms of each $S_{j}$ will be a large
number independent of $j$, actually $M_{j+1}-N_{j}=T$ for any $j$, while the
number of terms of the blocks $L_{j}$ may be arbitrarily large, that is,
$N_{j}-M_{j}$ may tend to $\infty$ as $j\to\infty$. However the proof is more
involved and several technical adjustments are needed. We have already
constructed $F_{1}=L_{1}$ and the arc $I_{1}$ satisfying (36) and (37) for
$k=1$. Assume, by induction, that the blocks $L_{1},\dotsc,L_{k}$,
$S_{1},\dotsc,S_{k-1}$ and the arcs $I_{k}\subset I_{k-1}\subset\dotsb\subset
I_{1}$ have been constructed verifying the induction assumptions. We will
complete the inductive step constructing the blocks $S_{k}$, $L_{k+1}$ and the
arc $I_{k+1}$.

Fix a constant $0<c_{1}=c_{1}(\eta)<1$ such that there exists
$z_{k}^{*}\in\mathbb{D}$ with $\rho(z_{k}^{*},z(I_{k}))\leq c_{1}$ such that
$\eta^{-1}I(z_{k}^{*})\subset I_{k}$. Observe that (36) and part (b) of Lemma
2.4 provide a constant $0<\gamma_{0}=\gamma_{0}(f)<1$ such that
$|f^{N_{k}}(z(I_{k}))|\leq\gamma_{0}$. By Schwarz’s Lemma there exists a
constant $0<\gamma_{1}=\gamma_{1}(\gamma_{0},\eta)<1$ such that
$|f^{N_{k}}(z_{k}^{*})|\leq\gamma_{1}$. Since $M_{k+1}=N_{k}+T$, estimate (33)
gives $|f^{M_{k+1}}(z_{k}^{*})|\leq\eta$. This defines $S_{k}$ and allows to
apply Corollary 3.1 to any partial sum starting at the index $M_{k+1}$. The
construction of the block $L_{k+1}$ and the arc $I_{k+1}$ requires to consider
two cases:

(I) Assume

(39) $\sum^{\infty}_{n=N_{k}+1}|a_{n}|\leq\frac{100d_{k}}{\eta}.$

Note that the induction hypothesis (37) gives

$\sum^{\infty}_{n=N_{k}+1}|a_{n}|\geq\frac{10d_{k}}{\eta}.$

Since $M_{k+1}=N_{k}+T$, estimate (34) gives

$\sum^{\infty}_{n=M_{k+1}}|a_{n}|\geq\frac{9d_{k}}{\eta}.$

Let $N_{k+1}$ be the smallest positive integer bigger than $M_{k+1}$ such that

$\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|\geq\frac{d_{k}\eta}{2}.$

This defines the block $L_{k+1}$. Note that

$\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|\leq\frac{d_{k}\eta}{2}+|a_{N_{k+1}}|.$

By estimate (35),

$|a_{N_{k+1}}|\leq\frac{\eta^{2}}{300}\sum\limits_{n>N_{k+1}}|a_{n}|.$

Hence estimate (39) gives

(40)
$\frac{5d_{k}\eta}{6}\geq\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|\geq\frac{d_{k}\eta}{2}.$

Let $\xi_{k}\in I_{k}$ such that $d_{k}=|w-F_{k}(\xi_{k})|$. Apply Corollary
3.1 to the point $z_{k}^{*}$, the value $w-F_{k}(\xi_{k})$ and the block
$L_{k+1}$, to find $\xi_{k+1}\in\eta^{-1}I(z_{k}^{*})\subset I_{k}$ such that

$|w-F_{k}(\xi_{k})-L_{k+1}(\xi_{k+1})|\leq
d_{k}-\eta\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|.$

Hence

(41) $|w-F_{k}(\xi_{k})-S_{k}(\xi_{k+1})-L_{k+1}(\xi_{k+1})|\leq
d_{k}-\eta\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|+\sum^{M_{k+1}-1}_{n=N_{k}+1}|a_{n}|.$

Since $m(f^{N_{k}}(I_{k}))=1/2$ and $f$ is expanding, one can find an arc
$I_{k+1}$ with $\xi_{k+1}\in I_{k+1}\subset I_{k}$ and
$m(f^{N_{k+1}}(I_{k+1}))=1/2$. We complete the inductive step showing

(42) $d_{k+1}\leq\frac{\eta}{10}\sum^{\infty}_{n=N_{k+1}+1}|a_{n}|.$

Let $K=\min\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\}$. Since $f(0)=0$
and $f$ is not a rotation, we have $K>1$. Note that by Lemma 2.1 we have

$|F_{k}(\xi_{k})-F_{k}(\xi_{k+1})|\leq\pi\sum^{N_{k}}_{n=1}|a_{n}|K^{n-N_{k}}.$

Hence (41) gives

$d_{k+1}\leq
d_{k}-\eta\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|+\sum^{M_{k+1}-1}_{n=N_{k}+1}|a_{n}|+\pi\sum^{N_{k}}_{n=1}|a_{n}|K^{n-N_{k}}.$

Applying the inductive assumption, the estimate (42) would follow from

$\frac{\eta}{10}\sum^{\infty}_{n=N_{k}+1}|a_{n}|-\eta\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|+\sum^{M_{k+1}-1}_{n=N_{k}+1}|a_{n}|+\pi\sum^{N_{k}}_{n=1}|a_{n}|K^{n-N_{k}}\leq\frac{\eta}{10}\sum^{\infty}_{n=N_{k+1}+1}|a_{n}|.$

Hence, we need to check

(43)
$\frac{-9\eta}{10}\sum^{N_{k+1}}_{n=M_{k+1}}|a_{n}|+\left(1+\frac{\eta}{10}\right)\sum^{M_{k+1}-1}_{n=N_{k}+1}|a_{n}|+\pi\sum^{N_{k}}_{n=1}|a_{n}|K^{n-N_{k}}\leq
0.$

Now, (40) says that the first sum is comparable to $d_{k}$. By Lemma 2.8 and
the assumption (39), last sum of (43) can be absorved by the fist one. On the
other hand $M_{k+1}=N_{k}+T$ and hence by (34), the second sum of (43) is also
absorved by the first one. We see that (43) holds and the inductive step is
completed in this case.

(II) Assume

(44) $\sum^{\infty}_{n=N_{k}+1}|a_{n}|\geq\frac{100d_{k}}{\eta}.$

Let $\xi_{k}\in I_{k}$ such that $d_{k}=|w-F_{k}(\xi_{k})|$ and pick
$N_{k+1}=M_{k+1}$, that is, in this case $L_{k+1}$ has a single term. Since
$m(f^{N_{k}}(I_{k}))=1/2$ and $f$ is expanding, one can take an arc $I_{k+1}$
with $\xi_{k}\in I_{k+1}\subset I_{k}$ and $m(f^{N_{k+1}}(I_{k+1}))=1/2$. Note
that

$d_{k+1}\leq d_{k}+\sum^{M_{k+1}-1}_{n=N_{k}+1}|a_{n}|+|a_{N_{k+1}}|.$

Applying (44) we deduce

$d_{k+1}\leq\frac{\eta}{100}\sum^{\infty}_{n=N_{k}+1}|a_{n}|+\sum^{M_{k+1}}_{n=N_{k}+1}|a_{n}|$

and to complete the inductive step, it is sufficient to check

$\frac{\eta}{100}\sum^{\infty}_{n=N_{k}+1}|a_{n}|+\sum^{M_{k+1}}_{n=N_{k}+1}|a_{n}|\leq\frac{\eta}{10}\sum^{\infty}_{n=N_{k+1}+1}|a_{n}|.$

The choice $M_{k+1}-N_{k}=T+1$ and estimate (34) finish the proof. ∎

##  4 Concluding remarks

Given $\xi\in\partial\mathbb{D}$ and $M>1$, let
$\Gamma(\xi,M)=\\{z\in\mathbb{D}:|z-\xi|\leq M(1-|z|)\\}$ be the Stolz angle
with vertex at $\xi$ and aperture depending on $M$. A function
$F:\mathbb{D}\rightarrow\mathbb{C}$ has a non tangential limit at the point
$\xi\in\partial\mathbb{D}$, which will be denoted by
$\lim_{{z\to\xi}\atop{\nprec}}F(z)$, if for any $M>1$ the limit

$\lim_{z\in\Gamma(\xi,M),z\to\xi}F(z)$

exists and is finite.

In the statements of Theorems 1.1 and 1.2 one can replace the sum
$\sum\limits_{n}a_{n}f^{n}(\xi)$ by the non-tangential limit

$\lim_{{z\to\xi}\atop{\nprec}}\sum_{n}a_{n}f^{n}(z).$

This follows from our next result which can be understood as a version of the
classical Abel’s Theorem in our context.

######  Theorem 4.1.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Let $\\{a_{n}\\}$ be a sequence of complex numbers and let
$\xi\in\partial\mathbb{D}$ such that $\sum_{n}a_{n}f^{n}(\xi)$ converges. Then
the non-tangential limit

$\lim_{{z\to\xi}\atop{\nprec}}\displaystyle{\sum_{n=1}^{\infty}}a_{n}f^{n}(z)$

exists and it is equal to $\sum\limits_{n=1}^{\infty}a_{n}f^{n}(\xi)$.

###### Proof.

Let $K_{1}=\max\\{|f^{\prime}(\xi)|\,:\,\xi\in\partial\mathbb{D}\\}$ and pick
$M>K_{1}$. Fix $0<\delta<(10K_{1})^{-1}$. For $z\in\Gamma(\xi,M)$ let $N(z)$
be the smallest positive integer such that $m(f^{N(z)}(I(z)))\geq\delta$.
Apply Lemma 2.6 to obtain a constant $C=C(\delta,f)>0$ such that

(45) $\sum^{N(z)}_{k=1}|f^{k}(\xi)-f^{k}(z)|\leq C,\quad\xi\in I(z).$

Note that $N(z)\to\infty$ as $|z|\to 1$ and $m(f^{N(z)}(I(z)))\leq\delta
K_{1}$. Part (b) of Lemma 2.4 gives a constant $0<\gamma=\gamma(\delta,f)<1$
such that $|f^{N(z)}(z)|\leq\gamma$. Pommerenke estimates of the Denjoy–Wolff
Theorem (Lemma 2 of [P]) provide constants $C_{1}=C_{1}(\gamma)>0$ and
$0<a=a(f)<1$ such that

(46) $|f^{n}(z)|\leq C_{1}a^{n-N(z)},\quad n\geq N(z).$

We are now ready to prove the result. Fix $\varepsilon>0$. We want to show
that there exists $N_{0}=N_{0}(\varepsilon)>0$ such that for any
$z\in\Gamma(\xi,M)$ sufficiently close to $\xi$, we have

(47)
$\left|\sum^{\infty}_{n=N}a_{n}(f^{n}(z)-f^{n}(\xi))\right|\leq\varepsilon,\quad
N\geq N_{0}.$

Summing by parts

$\sum^{\infty}_{n=N}a_{n}(f^{n}(z)-f^{n}(\xi))=\sum^{\infty}_{n=N+1}S_{n}(\xi)\left(\frac{f^{n}(z)}{f^{n}(\xi)}-\frac{f^{n-1}(z)}{f^{n-1}(\xi)}\right)+S_{N}(\xi)\left(\frac{f^{N}(z)}{f^{N}(\xi)}-1\right),$

where

$S_{n}(\xi)=\sum^{\infty}_{k=n}a_{k}f^{k}(\xi).$

Next, we will show that there exists a constant $C_{2}=C_{2}(\delta,f)>0$ such
that

(48)
$\sum_{n=1}^{\infty}\left|\frac{f^{n}(z)}{f^{n}(\xi)}-\frac{f^{n-1}(z)}{f^{n-1}(\xi)}\right|\leq
C_{2}.$

Note that estimate (46) gives a constant $C_{3}=C_{3}(\delta,f)>0$ such that,

$\sum_{n=1}^{\infty}\left|\frac{f^{n}(z)}{f^{n}(\xi)}-\frac{f^{n-1}(z)}{f^{n-1}(\xi)}\right|\leq\sum_{n=1}^{N(z)}\left|f^{n}(z)f^{n-1}(\xi)-f^{n-1}(z)f^{n-1}(\xi)\right|+C_{3}.$

Adding and substracting ${f^{n}(\xi)}{f^{n-1}(\xi)}$ and applying estimate
(45), we obtain (48). By assumption, we can choose $N_{0}$ such that
$|S_{n}(\xi)|\leq\varepsilon$ if $n\geq N_{0}$. This gives (47) and finishes
the proof. ∎

Another turn of the screw of the methods in the proof of Theorem 1.1 leads to
our next result which is formally, a generalization of Theorem 1.1. An
analogous result in the context of lacunary series can be found in [KWW].
Given a sequence of complex numbers $\\{b_{n}\\}$ let $\\{\sum
b_{n}\\}^{\prime}$ denote the set of accumulation points of its partial sums,
that is,

$\\{\sum
b_{n}\\}^{\prime}=\bigcap_{n=1}^{\infty}\overline{\left\\{\sum_{k=1}^{m}b_{k}\,:\,m\geq
n\right\\}}.$

Note that if $\\{b_{n}\\}$ tends to $0$, then $\\{\sum b_{n}\\}^{\prime}$ is a
closed connected set of the extended complex plane.

######  Theorem 4.2.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Let $\\{a_{n}\\}$ be a sequence of complex numbers tending to $0$ such that
$\sum|a_{n}|=\infty$. Then, for any closed connected set $K$ of the extended
complex plane, there exists a point $\xi\in\partial\mathbb{D}$ such that

$K=\\{\sum a_{n}f^{n}(\xi)\\}^{\prime}.$

###### Proof of Theorem 4.2.

Since the proof only requires an iteration of the ideas of the proof of
Theorem 1.1, we will only sketch it. Given the set $K$, let us consider a
sequence $\\{w_{n}\\}$ of complex numbers with $|w_{n+1}-w_{n}|\to 0$ as
$n\to\infty$, such that

$K=\bigcap_{n=1}^{\infty}\overline{\\{w_{m}\,:\,m\geq n\\}}.$

We use the notation $S_{N}(\xi)=\sum_{n=1}^{N}a_{n}f^{n}(\xi)$. Using the
methods in the proof of Theorem 1.1 one can construct a closed arc
$I_{1}\subset\partial\mathbb{D}$ and an integer $N_{1}$ so that for any
$\xi\in I_{1}$, the partial sum $S_{N_{1}}(\xi)$ is close to $w_{1}$ and
$S_{m}(\xi)$ is not far from the segment $[0,w_{1}]$ for any $0<m\leq N_{1}$.
Once one has constructed this arc, using again the same argument, one
constructs a closed subarc $I_{2}\subset I_{1}$ and an integer $N_{2}>N_{1}$
such that for any $\xi\in I_{2}$, the partial sum $S_{N_{2}}(\xi)$ is close to
$w_{2}$ and $S_{m}(\xi)$ is not far from the segment $[w_{1},w_{2}]$ for any
$N_{1}\leq m\leq N_{2}$. Iterating this process and taking $\xi\in\bigcap
I_{j}$ one obtains the desired result. ∎

Theorem 4.2 has the corresponding counterpart in the context of radial
behaviour of the function $\displaystyle\sum a_{n}f^{n}$. Given an analytic
function $g:\mathbb{D}\to\mathbb{C}$, the radial cluster set
$\text{Cl}_{r}(g,\xi)$ of the function $g$ at the point
$\xi\in\partial\mathbb{D}$ is defined to be

$\text{Cl}_{r}(g,\xi)=\bigcap_{r<1}\overline{\left\\{g(s\xi)\,:\,s\geq
r\right\\}}.$

We will need the following easy consequence of Lemma 2.6.

######  Lemma 4.3.

Let $f$ be a finite Blaschke product with $f(0)=0$ which is not a rotation.
Fix $0<\delta<(10\max\\{|f^{\prime}(\xi)|:\xi\in\partial\mathbb{D}\\})^{-1}$.
Then there exits a constant $C=C(\delta,f)>0$ such that the following
statements hold.

(a) Given $z\in\mathbb{D}$ let $N(z)$ be the smallest positive integer such
that $m(f^{N(z)}(I(z)))\geq\delta$. Then

$\left|\sum_{n=1}^{\infty}a_{n}f^{n}(z)-\sum_{n=1}^{N(z)}a_{n}f^{n}(\xi)\right|\leq
C\sup_{n}|a_{n}|,\quad\xi\in I(z).$

(b) Fix $\xi\in\partial\mathbb{D}$. Given a positive integer $N$, consider
$r_{N}=\sup\\{0<r<1\ :m(f^{N}(I(r\xi)))\geq\delta\\}$. Then

$\left|\sum_{n=1}^{N}a_{n}f^{n}(\xi)-\sum_{n=1}^{\infty}a_{n}f^{n}(r_{N}\xi)\right|\leq
C\sup_{n}|a_{n}|.$

###### Proof.

Apply Lemma 2.6 to obtain a constant $C=C(\delta,f)>0$ such that

$\sum^{N(z)}_{n=1}|a_{n}||f^{n}(\xi)-f^{n}(z)|\leq
C\sup_{n}|a_{n}|,\quad\xi\in I(z).$

The exponential decay (46) given by Pommerenke’s result, provides a constant
$C_{1}=C_{1}(\delta)>0$ such that

$\sum_{n=N(z)}^{\infty}|f^{n}(z)|\leq C_{1}.$

This finishes the proof of (a). Part (b) follows similarly. ∎

Let $\\{a_{n}\\}$ be a sequence of complex numbers tending to $0$ and $F=\sum
a_{n}f^{n}$. Note that in the previous Lemma we have that $N(z)\to\infty$ as
$|z|\to 1$ and $r_{N}\to 1$ as $N\to\infty$. We deduce that
$\text{Cl}_{r}(F,\xi)=\\{\sum a_{n}f^{n}(\xi)\\}^{\prime}$ for any
$\xi\in\partial\mathbb{D}$. Hence next result follows from Theorem 4.2.

######  Theorem 4.4.

Let $f$ and $\\{a_{n}\\}$ be as in Theorem 4.2 and let $F$ be the analytic
function defined by $\displaystyle F(z)=\sum_{n=1}^{\infty}a_{n}f^{n}(z)$,
$z\in\mathbb{D}$. Then, for any closed connected set $K$ of the extended
complex plane there exists a point $\xi\in\partial\mathbb{D}$ such that
$Cl_{r}(F,\xi)=K.$

We finally mention some related open problems we have not explored.

  1. 1.

Let the function $f$ and the coefficients $\\{a_{n}\\}$ be as in the statement
of Theorem 1.1. Fix $w\in\mathbb{C}$. It is natural to study the size of the
set $E(w)$ of points $\xi\in\partial\mathbb{D}$ such that $\sum
a_{n}f^{n}(\xi)=w$. Note that if we assume $\sum|a_{n}|^{2}=\infty$, then
$\sum a_{n}f^{n}$ is a function in the Bloch space that has radial limit at
almost no point of the unit circle (see [N]). In this situation Rohde proved
that the set

$\left\\{\xi\in\partial\mathbb{D}:\lim_{r\to
1}\sum_{n=1}^{\infty}a_{n}f^{n}(r\xi)=w\right\\}$

has Hausdorff dimension $1$. See [Ro]. It is also natural to ask for the size
of the set of points $\xi\in\partial\mathbb{D}$ where the conclusion of
Theorem 4.2 holds.

  2. 2.

Let $F(z)=\sum a_{n}z^{k_{n}}$ be a lacunary power series with radius of
convergence equals to $1$, such that $\sum|a_{n}|=\infty$. Murai proved that
for any $w\in\mathbb{C}$ there are infinitely many $z\in\mathbb{D}$ such that
$F(z)=w$ ([Mu3]). It is natural to seek for analogus results in our context.
More concretely, let $f$ and $\\{a_{n}\\}$ be as in the statement of Theorem
1.1. Is it true that for any $w\in\mathbb{C}$ there exists $z\in\mathbb{D}$
such that $\sum a_{n}f^{n}(z)=w$?

  3. 3.

The High Indices Theorem of Hardy and Littlewood provides a converse of Abel’s
Theorem in the setting of lacunary series. See [K]. Hence it is natural to ask
for the converse of Theorem 4.1.

## References

  * [Ba] Barański, K. On some lacunary power series. _Michigan Math. J._ 54 (2006), no. 1, 65–79.
  * [Be] Belov, A. S. On the Salem and Zygmund problem with respect to the smoothness of an analytic function that generates a Peano curve. (Russian) _Mat. Sb._ 181 (1990), no. 8, 1048–1060; translation in _Math. USSR-Sb._ 70 (1991), no. 2, 485–497.
  * [KWW] Kahane, J.-P.; Weiss, M.; Weiss, G. On lacunary power series. _Ark. Mat._ 5 (1963), 1–26.
  * [K] Korevaar, J. _Tauberian theory. A century of developments_. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 329. Springer-Verlag, Berlin, 2004.
  * [Mu1] Murai, T. The value-distribution of lacunary series and a conjecture of Paley. _Ann. Inst. Fourier (Grenoble)_ 31 (1981), no. 1, vii, 135–156.
  * [Mu2] Murai, T. On lacunary series. _Nagoya Math. J._ 85 (1982), 87–154.
  * [Mu3] Murai, T. The boundary behaviour of Hadamard lacunary series. _Nagoya Math. J._ 89 (1983), 65–76.
  * [N] Nicolau, A. Convergence of linear combinations of iterates of an inner function. Preprint, 2021.
  * [NS] Nicolau, A.; Soler i Gibert, O. A central limit theorem for inner functions. Preprint, 2020.
  * [P] Pommerenke, Ch. On ergodic properties of inner functions. _Math. Ann._ 256 (1981), no. 1, 43–50.
  * [Ro] Rohde, S. The boundary behavior of Bloch functions. _J. London Math. Soc._ 48 (1993), 488–499.
  * [SZ] Salem, R.; Zygmund, A. Lacunary power series and Peano curves. _Duke Math. J._ 12 (1945), 569–578.
  * [W] Weiss, M. Concerning a theorem of Paley on lacunary power series. _Acta Math._ 102 (1959), 225–238.
  * [Y] Younsi, M. Peano curves in complex analysis. _Amer. Math. Monthly_ 126 (2019), no. 7, 635–640.
  * [Z] Zygmund, A. _Trigonometric series_. Vol. I, II. Second edition, reprinted with corrections and some additions. Cambridge University Press, London-New York, 1968.


# Euler characteristic of the space of
real multivariate irreducible polynomials

Trevor Hyde  Dept. of Mathematics
University of Chicago
Chicago, IL 60637
<EMAIL_ADDRESS>
Given a field $K$ and integers $d,n\geq 1$, let $\mathrm{Poly}_{d,n}(K)$
denote the set of all monic total degree $d$ polynomials in $n$ variables with
coefficients in $K$ (see Definition 2.1.) Let
$\mathrm{Irr}_{d,n}(K)\subseteq\mathrm{Poly}_{d,n}(K)$ denote the subset of
all polynomials which are irreducible over $K$. Our main result expresses the
compactly supported Euler characteristic of $\mathrm{Irr}_{d,n}(\mathbb{R})$
in terms of the so-called balanced binary expansion of the number of variables
$n$. The balanced binary expansion of an integer $n\geq 1$ is the unique
expression

$n=\sum_{k=0}^{\ell}b_{k}2^{k}=2^{k_{2m}}-2^{k_{2m-1}}+2^{k_{2m-2}}-\ldots+2^{k_{1}}-2^{k_{0}}$

where $0\leq k_{0}<k_{1}<\ldots<k_{2m}$ is a strictly increasing sequence of
natural numbers of even length and the signs on the right hand side alternate.

######  Theorem 1.1.

Let $n\geq 1$ be an integer with balanced binary expansion
$n=\sum_{k=0}^{\ell}b_{k}2^{k}$ and let $\chi_{c}$ denote the compactly
supported Euler characteristic. Then

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=\begin{cases}b_{k}&\text{if
}d=2^{k},\\\ 0&\text{otherwise.}\end{cases}$

######  Example 1.2.

The balanced binary expansion of $n=13$ is

$13=2^{4}-2^{2}+2^{1}-1.$

Thus Theorem 1.1 implies that

$\chi_{c}(\mathrm{Irr}_{d,13}(\mathbb{R}))=\begin{cases}\hskip 7.5pt1&\text{if
}d=2\text{ or }16,\\\ -1&\text{if }d=1\text{ or }4,\\\ \hskip
7.5pt0&\text{otherwise.}\end{cases}$

That is, the space of degree $d$ irreducible polynomials in 13 variables with
coefficients in $\mathbb{R}$ has vanishing compactly supported Euler
characteristic in all degrees except $d=1,2,4,16$, and these exceptional
degrees are determined by the binary expansion of $n=13$.

######  Example 1.3.

For univariate polynomials ($n=1$) we can explicitly construct cell
decompositions of $\mathrm{Irr}_{d,1}(\mathbb{R})$ for all $d\geq 1$. In
particular, the fundamental theorem of algebra and the quadratic formula imply

$\displaystyle\mathrm{Irr}_{1,1}(\mathbb{R})$
$\displaystyle=\\{x+a:a\in\mathbb{R}\\}\cong\mathbb{R},$
$\displaystyle\mathrm{Irr}_{2,1}(\mathbb{R})$
$\displaystyle=\\{x^{2}+bx+c:b^{2}<4c\\}\cong\mathbb{R}^{2},$
$\displaystyle\mathrm{Irr}_{d,1}(\mathbb{R})$ $\displaystyle=\emptyset\text{
for $d>2$.}$

Since $\chi_{c}(\mathbb{R}^{k})=(-1)^{k}$ it follows that

$\chi_{c}(\mathrm{Irr}_{d,1}(\mathbb{R}))=\begin{cases}-1&\text{if }d=1,\\\
\hskip 7.5pt1&\text{if }d=2,\\\ \hskip 7.5pt0&\text{if }d>2.\end{cases}$

This calculation is consistent with Theorem 1.1 since $1=2-1$ is the balanced
binary expansion of $1$.

The connection between the Euler characteristic of
$\mathrm{Irr}_{d,n}(\mathbb{R})$ and the binary expansion of the number of
variables $n$ was discovered empirically and came as a surprise to the author.
This result suggests that the spaces $\mathrm{Irr}_{d,n}(\mathbb{R})$ may have
interesting cell decompositions determined by the additive structure of $n$.
Our proof of Theorem 1.1 arrives at this Euler characteristic computation
indirectly via generating functions; the dependence on the binary expansion of
$n$ passes through the well-known theorem of Lucas on the mod $p$ residue of
binomial coefficients (see Lemma 4.6.)

Chen [1] proved that the singular and compactly supported cohomology of the
spaces $\mathrm{Irr}_{d,n}(\mathbb{C})$ stabilizes in certain cohomological
degree regimes as $n\rightarrow\infty$. We do not know whether Chen’s methods
can be adapted to analyze the compactly supported cohomology of
$\mathrm{Irr}_{d,n}(\mathbb{R})$.

In [6] the author introduced the higher necklace polynomials
$M_{d,n}(x)\in\mathbb{Q}[x]$ which interpolate the point counts of
$\mathrm{Irr}_{d,n}(\mathbb{F}_{q})$ for finite fields $\mathbb{F}_{q}$ where
$q$ is a prime power,

$|\mathrm{Irr}_{d,n}(\mathbb{F}_{q})|=M_{d,n}(q).$

En route to proving Theorem 1.1 we show that the compactly supported Euler
characteristics of $\mathrm{Irr}_{d,n}(\mathbb{R})$ and
$\mathrm{Irr}_{d,n}(\mathbb{C})$ are realized as values of the higher necklace
polynomials.

######  Theorem 1.4.

For all $d,n\geq 1$,

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=M_{d,n}(-1)\hskip
54.2025pt\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{C}))=M_{d,n}(1)=\begin{cases}n&\text{if
}d=1\\\ 0&\text{otherwise.}\end{cases}$

The compactly supported Euler characteristic may be viewed as a topological
extension of the cardinality of a finite set. Since $\chi_{c}(\mathbb{R})=-1$
and $\chi_{c}(\mathbb{C})=1$, it is sometimes said that $\mathbb{R}$ and
$\mathbb{C}$ behave like “fields with $-1$ and $1$ element” respectively;
Theorem 1.4 provides an example of a common situation where this heuristic
applies. The strategy behind Theorem 1.4 and similar results is to find some
universal decomposition of $\mathrm{Irr}_{d,n}(K)$ into a disjoint union of
affine cells $K^{m}$. Provided the decomposition is sufficiently nice,
algebraic properties of point counts and compactly supported Euler
characteristics translate these universal cell decompositions into a common
polynomial expression for the respective invariants.

With Theorem 1.4 in hand, Theorem 1.1 reduces to evaluating the polynomial
$M_{d,n}(x)$ at $x=-1$. More generally we prove the following result on
evaluations $M_{d,n}(\zeta_{p})$ of the higher necklace polynomials at prime
order roots of unity. Note that a balanced base $p$ expansion of an integer
$n$ is an expression

$n=\sum_{k=0}^{\ell}b_{k}p^{k}=p^{k_{2m}}-p^{k_{2m-1}}+p^{k_{2m-2}}-\ldots+p^{k_{1}}-p^{k_{0}}$

where $0\leq k_{0}<k_{1}<\ldots<k_{2m}$ is a strictly increasing sequence of
natural numbers of even length and the signs on the right hand side alternate
(see Section 4.)

######  Theorem 1.5.

Let $p$ be a prime, let $d,n\geq 1$ be integers, and suppose that $n$ has a
balanced base $p$ expansion $n=\sum_{k=0}^{\ell}b_{k}p^{k}$. If $\zeta_{p}$ is
a primitive $p$th root of unity, then

$M_{d,n}(\zeta_{p})=\begin{cases}b_{k}&\text{if }d=p^{k}\\\
0&\text{otherwise.}\end{cases}$

In particular, for a fixed $n$ with balanced base $p$ expansion, $M_{d,n}(x)$
is divisible by the $p$th cyclotomic polynomial $\Phi_{p}(x)$ for all but
finitely many $d$. When $n=1$ the higher necklace polynomials specialize to
the classic sequence $M_{d}(x):=M_{d,1}(x)$ of necklace polynomials. Odesky
and the author [9] observed that the necklace polynomials $M_{d}(x)$ are
highly reducible over $\mathbb{Q}$ with nearly all of their irreducible
factors being cyclotomic polynomials. They explain this phenomenon in terms of
the representation theory of finite abelian groups and the combinatorics of
hyperplane arrangements. While their theory only applies when $n=1$, Theorem
1.5 implies that the phenomenon of unexpected cyclotomic factors extends,
albeit partially, to the higher necklace polynomials $M_{d,n}(x)$ as well.

Finally we note the generating function identity which plays a critical role
in our analysis and may be of independent interest. Let
$P_{d,n}(x)\in\mathbb{Q}[x]$ denote the polynomial (see Section 2) such that
for all prime powers $q$,

$|\mathrm{Poly}_{d,n}(\mathbb{F}_{q})|=P_{d,n}(q).$

######  Theorem 1.6 (Higher Cyclotomic Identity).

For each $n\geq 1$, the following identity holds in
$\mathbb{Q}[x][\\![t]\\!]$,

$\sum_{d\geq 0}P_{d,n}(x)t^{d}=\prod_{j\geq
1}\Big{(}\frac{1}{1-t^{j}}\Big{)}^{M_{j,n}(x)}.$

When $n=1$, $P_{d,1}(x)=x^{d}$ and Theorem 1.6 specializes to the classic so-
called cyclotomic identity,

$\frac{1}{1-xt}=\sum_{d\geq 0}x^{d}t^{d}=\prod_{d\geq
1}\Big{(}\frac{1}{1-t^{d}}\Big{)}^{M_{d}(x)}.$

There are several interpretations of the cyclotomic identity, including with
$x=q$ as an Euler product for the Hasse-Weil zeta function
$\zeta_{\mathbb{A}^{1}(\mathbb{F}_{q})}(t)$ of the affine line.

###  1.1. Organization

The body of this paper is divided into four sections. In Section 2 we prove
the higher cyclotomic identity. In Section 3 we realize the compactly
supported Euler characteristics of $\mathrm{Irr}_{d,n}(\mathbb{R})$ and
$\mathrm{Irr}_{d,n}(\mathbb{C})$ as values of the higher necklace polynomials.
In Section 4 we evaluate the higher necklace polynomials at prime order $p$
roots of unity in terms of the balanced base $p$ expansion of the number of
variables $n$. Finally, in Section 5 we bring everything together to express
the compactly supported Euler characteristic of
$\mathrm{Irr}_{d,n}(\mathbb{R})$ in terms of the balanced binary expansion of
$n$.

###  1.2. Note on previous versions

The results in this paper originally appeared in the author’s Ph.D.
dissertation [8] and as the second half of the preprint [7]. This earlier
preprint has split into two papers with a coauthor added to the first half
[9].

###  1.3. Acknowledgements

We would like to thank Benson Farb, Nir Gadish, and Phil Tosteson for helpful
discussions related to this work. The author is partially supported by the NSF
MSPRF and the Jump Trading Mathlab Research Fund.

##  2\. Higher necklace polynomials

In this section we prove the higher cyclotomic identity. First we recall the
definitions of the sets $\mathrm{Poly}_{d,n}(K)$, $\mathrm{Irr}_{d,n}(K)$ and
their associated polynomials $P_{d,n}(x)$, $M_{d,n}(x)$ from [6]. Let $d,n\geq
1$ be integers and let $K$ be an arbitrary field.

######  Definition 2.1.

Let $\mathrm{Poly}_{d,n}(K)$ be the set of $K^{\times}$-orbits of total degree
$d$ polynomials in $K[x_{1},x_{2},\ldots,x_{n}]$. The total degree of a
monomial $x_{1}^{m_{1}}x_{2}^{m_{2}}\cdots x_{n}^{m_{n}}$ is
$m_{1}+m_{2}+\ldots+m_{n}$ and the total degree of a polynomial $f\in
K[x_{1},x_{2},\ldots,x_{n}]$ is the maximum total degree of the monomial
summands of $f$. We refer to the elements of $\mathrm{Poly}_{d,n}(K)$ as monic
polynomials. Let $\mathrm{Irr}_{d,n}(K)\subseteq\mathrm{Poly}_{d,n}(K)$ be the
subset of all monic polynomials which are irreducible over $K$.

Note that if we choose a monomial ordering on $K[x_{1},x_{2},\ldots,x_{n}]$,
then there is a unique representative for each $K^{\times}$-orbit with leading
coefficient 1. Hence our notion of a monic polynomial is equivalent to the
more common notion after such a choice of monomial order.

Recall that for any field $K$, the polynomial ring
$K[x_{1},x_{2},\ldots,x_{n}]$ is a unique factorization domain. Unique
factorization gives us the decomposition

$\mathrm{Poly}_{d,n}(K)=\bigsqcup_{\lambda\vdash d}\prod_{j\geq
1}\mathrm{Sym}^{m_{j}(\lambda)}(\mathrm{Irr}_{j,n}(K)),$ (2.1)

where the disjoint union is over all partitions $\lambda$ of $d$ and
$m_{j}(\lambda)$ is the number of parts of $\lambda$ of size $j$. Here
$\mathrm{Sym}^{m}(X):=X^{m}/S_{m}$ denotes the $m$-fold symmetric product.

In [6, Lem. 2.1] the author proved that there are polynomials
$P_{d,n}(x),M_{d,n}(x)\in\mathbb{Q}[x]$ such that for any finite field
$\mathbb{F}_{q}$,

$|\mathrm{Poly}_{d,n}(\mathbb{F}_{q})|=P_{d,n}(q)\hskip
54.2025pt|\mathrm{Irr}_{d,n}(\mathbb{F}_{q})|=M_{d,n}(q).$

We call $M_{d,n}(x)$ the higher necklace polynomials. Let
$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{m}\right)\kern-3.00003pt\right)\in\mathbb{Q}[x]$
be the polynomial

$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{m}\right)\kern-3.00003pt\right):=\frac{x(x+1)(x+2)\cdots(x+m-1)}{m!}=\binom{x+m-1}{m}.$
(2.2)

If $n\in\mathbb{N}$, then
$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{n}{m}\right)\kern-3.00003pt\right)$
may be interpreted as the number of ways to choose $m$ objects from a
collection of $n$ objects with repetitions. Taking cardinalities of both sides
in (2.1) with $K=\mathbb{F}_{q}$ gives us the polynomial identity

$P_{d,n}(x)=\sum_{\lambda\vdash d}\prod_{j\geq
1}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{M_{j,n}(x)}{m_{j}(\lambda)}\right)\kern-3.00003pt\right).$
(2.3)

######  Definition 2.2.

A commutative ring $R$ is called a binomial ring if

  1. (1)

$R$ is torsion free as an abelian group ($ma=0$ with $m\in\mathbb{Z}$ and
$a\in A$ implies $m=0$ or $a=0$,)

  2. (2)

For each $a\in R$ and $n\geq 0$,
$\binom{a}{n}=\frac{1}{n!}a(a-1)(a-2)\cdots(a-n+1)\in R$.

Binomial rings were defined by Philip Hall [4] in his study of nilpotent
groups. See Elliott [2] for an overview and further references on binomial
rings. Examples of binomial rings include any localization of $\mathbb{Z}$,
any $\mathbb{Q}$-algebra, and the ring of integer valued polynomials in
$\mathbb{Q}[x]$. The second condition in Definition 2.2 is equivalent to
$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{a}{n}\right)\kern-3.00003pt\right)\in
R$ for all $a\in R$ and $n\geq 0$ as can be seen by (2.2).

Let $R$ be a binomial ring and let $\Lambda(R):=1+tR[\\![t]\\!]$ be the
multiplicative group of unital formal power series with coefficients in $R$.
We use
$\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{x}{n}\right)\kern-3.00003pt\right)$
to define an exponential action of $R$ on certain elements of $\Lambda(R)$. In
particular,

$\left(\frac{1}{1-t}\right)^{a}:=\sum_{n\geq
0}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{a}{n}\right)\kern-3.00003pt\right)t^{n}.$

This identity is equivalent to the binomial theorem by (2.2).

Lemma 2.3 is well-known in the context of formal power series, symmetric
functions, and the theory of Witt vectors but is typically not stated in the
generality which we technically require. As we will make use of this several
times we prove it here for completeness.

######  Lemma 2.3.

For any binomial ring $R$ and any sequence $a_{d}\in R$ for $d\geq 0$ such
that $a_{0}=1$ there exists a unique sequence $b_{j}\in R$ for $j\geq 1$ such
that the following identity holds in $\Lambda(R)$,

$\sum_{d\geq 0}a_{d}t^{d}=\prod_{j\geq
1}\left(\frac{1}{1-t^{j}}\right)^{b_{j}}.$ (2.4)

Furthermore (2.4) is equivalent to

$a_{d}=\sum_{\lambda\vdash d}b_{\lambda}$

for all $d\geq 1$ where for a partition $\lambda=(1^{m_{1}}2^{m_{2}}\cdots)$

$b_{\lambda}:=\prod_{j\geq
1}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{b_{j}}{m_{j}}\right)\kern-3.00003pt\right).$
(2.5)

###### Proof.

The right hand side of (2.4) expands as

$\prod_{j\geq 1}\left(\frac{1}{1-t^{j}}\right)^{b_{j}}=\prod_{j\geq
1}\sum_{m\geq
0}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{b_{j}}{m}\right)\kern-3.00003pt\right)t^{mj}=\sum_{d\geq
0}\sum_{\lambda\vdash d}b_{\lambda}t^{d}.$

We show by induction on $d$ that there exists a uniquely determined sequence
$b_{j}$ such that for all $d\geq 1$,

$a_{d}=\sum_{\lambda\vdash d}b_{\lambda}.$

For $d=1$ there is only partition $\lambda$ and thus $a_{1}=b_{1}$. Now
suppose that $d>1$ and that we have shown $b_{j}$ is uniquely determined for
$j<d$. Then

$b_{d}=a_{d}-\sum_{\begin{subarray}{c}\lambda\vdash d\\\
\lambda\neq(d)\end{subarray}}b_{\lambda}.$

If $\lambda\neq(d)$, then all parts of $\lambda$ have size $j<d$ hence $b_{d}$
is uniquely determined by our induction hypothesis. ∎

######  Theorem 2.4 (Higher Cyclotomic Identity).

For each $n\geq 1$, the following identity holds in $\Lambda(\mathbb{Q}[x])$,

$\sum_{d\geq 0}P_{d,n}(x)t^{d}=\prod_{j\geq
1}\Big{(}\frac{1}{1-t^{j}}\Big{)}^{M_{j,n}(x)}.$

###### Proof.

This follows directly from Lemma 2.3 and the identity (2.3) with
$R=\mathbb{Q}[x]$, $a_{d}=P_{d,n}(x)$, and $b_{j}=M_{j,n}(x)$. ∎

##  3\. Euler characteristics as higher necklace values

The goal of this section is prove that the compactly supported Euler
characteristics of $\mathrm{Irr}_{d,n}(\mathbb{R})$ and
$\mathrm{Irr}_{d,n}(\mathbb{C})$ may be realized as values of the higher
necklace polynomials. We begin with some background on compactly supported
Euler characteristics.

######  Definition 3.1.

Say a topological space $X$ is tame if the compactly supported singular
cohomology $H_{c}^{k}(X,\mathbb{Q})$ (see Hatcher [5, Pg. 243]) vanishes for
all but finitely many $k$. If $X$ is tame, then the compactly supported Euler
characteristic $\chi_{c}(X)$ is

$\chi_{c}(X):=\sum_{k\geq 0}(-1)^{k}\dim_{\mathbb{Q}}H_{c}^{k}(X,\mathbb{Q}).$

Affine and projective spaces over $\mathbb{R}$ and $\mathbb{C}$ are tame as
are locally closed algebraic subsets of projective spaces and their images
under algebraic maps. Tame spaces are closed under disjoint unions and
Cartesian products. Lemma 3.2 implies that $\mathrm{Poly}_{d,n}(\mathbb{R})$
and $\mathrm{Poly}_{d,n}(\mathbb{C})$ are tame. It then follows by induction
from (2.1) that $\mathrm{Irr}_{d,n}(\mathbb{R})$ and
$\mathrm{Irr}_{d,n}(\mathbb{C})$ are tame for all $d,n\geq 1$.

######  Lemma 3.2.

The set $\mathrm{Poly}_{\leq d,n}(K)$ of all monic polynomials with total
degree at most $d$ is naturally in bijection with the projective space
$\mathbb{P}^{\binom{d+n}{n}-1}(K)$, and thus with respect to the natural
inclusion $\mathrm{Poly}_{\leq d-1,n}(K)\subseteq\mathrm{Poly}_{\leq d,n}(K)$
we can identify $\mathrm{Poly}_{d,n}(K)$ with the complement

$\mathrm{Poly}_{d,n}(K)\cong\mathbb{P}^{\binom{d+n}{n}-1}(K)\setminus\mathbb{P}^{\binom{d+n-1}{n}-1}(K).$

###### Proof.

Consider the $K$-vector space spanned by all monomials in $n$ variables of
degree at most $d$. A standard counting argument implies that this space has
dimension $\binom{d+n}{n}$. The projectivization of this vector space is, by
definition, the space of all non-zero monic degree at most $d$ polynomials in
$K[x_{1},x_{2},\ldots,x_{n}]$. Hence $\mathrm{Poly}_{\leq
d,n}(K)\cong\mathbb{P}^{\binom{d+n}{n}-1}(K)$. ∎

Lemma 3.3 recalls some of the facts about $\chi_{c}$ that we will need.

######  Lemma 3.3.

Suppose that $X$ and $Y$ are tame spaces. Then

  1. (1)

$\chi_{c}(X\sqcup Y)=\chi_{c}(X)+\chi_{c}(Y)$,

  2. (2)

$\chi_{c}(X\times Y)=\chi_{c}(X)\chi_{c}(Y)$,

  3. (3)

$\chi_{c}(\mathbb{R})=-1$ and $\chi_{c}(\mathbb{C})=1$,

  4. (4)

If $K=\mathbb{R}$ or $\mathbb{C}$, then
$\chi_{c}(\mathbb{P}^{n-1}(K))=[n]_{\chi_{c}(K)}$, where

$[n]_{x}:=\frac{x^{n}-1}{x-1}.$

  5. (5)

$\mathrm{Sym}^{m}X$ is tame for all $m\geq 1$ and

$\chi_{c}(\mathrm{Sym}^{m}X)=\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{\chi_{c}(X)}{m}\right)\kern-3.00003pt\right).$

Equivalently, in $\Lambda(\mathbb{Z})=1+t\mathbb{Z}[\\![t]\\!]$ we have

$\sum_{d\geq
0}\chi_{c}(\mathrm{Sym}^{d}X)t^{d}=\left(\frac{1}{1-t}\right)^{\chi_{c}(X)}.$

###### Proof.

Part (1) is a general property of Euler characteristics and (2) holds for the
compactly supported Euler characteristic (see [12, Thm. 9.3.1].) To see (3) we
begin with the homeomorphism

$\mathbb{R}=(-\infty,0)\sqcup\\{0\\}\sqcup(0,\infty)\cong\mathbb{R}\sqcup\\{0\\}\sqcup\mathbb{R},$

and take $\chi_{c}$ of both sides to get

$\chi_{c}(\mathbb{R})=2\chi_{c}(\mathbb{R})+1\Longrightarrow\chi_{c}(\mathbb{R})=-1.$

Then $\mathbb{C}\cong\mathbb{R}^{2}$ and (2) imply
$\chi_{c}(\mathbb{C})=\chi_{c}(\mathbb{R})^{2}=1$. To compute the Euler
characteristic of projective space we use the cell decomposition

$\mathbb{P}^{n-1}(K)=K^{n-1}\sqcup K^{n-2}\sqcup\ldots\sqcup K\sqcup 1,$

where $1:=K^{0}$ is the one point space. Taking $\chi_{c}$ when $K=\mathbb{R}$
or $\mathbb{C}$ we have

$\chi_{c}(\mathbb{P}^{n-1}(K))=\chi_{c}(K)^{n-1}+\chi_{c}(K)^{n-2}+\ldots+\chi_{c}(K)+1=[n]_{\chi_{c}(K)}.$

The final assertion (5) is a theorem due to MacDonald [11]; see Vakil’s notes
[13, Thm. 2.3] for a nice one line proof. ∎

Lemma 3.2 and the cell decomposition of projective space used in the proof of
Lemma 3.3 together imply that

$P_{d,n}(x)=\left[\binom{n+d}{n}\right]_{x}-\left[\binom{n+d-1}{n}\right]_{x}.$
(3.1)

We now prove the main result of this section.

######  Theorem 3.4.

If $d,n\geq 1$, then

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=M_{d,n}(-1)\hskip
54.2025pt\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{C}))=M_{d,n}(1).$

###### Proof.

Let $K=\mathbb{R}$ or $\mathbb{C}$. Then (2.1) and Lemma 3.3 imply that

$\chi_{c}(\mathrm{Poly}_{d,n}(K))=\sum_{\lambda\vdash d}\prod_{j\geq
1}\chi_{c}(\mathrm{Sym}^{m_{j}}(\mathrm{Irr}_{j,n}(K)))=\sum_{\lambda\vdash
d}\prod_{j\geq
1}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{\chi_{c}(\mathrm{Irr}_{j,n}(K))}{m_{j}}\right)\kern-3.00003pt\right).$

Lemma 2.3 implies that the above identity is equivalent to

$\sum_{d\geq 0}\chi_{c}(\mathrm{Poly}_{d,n}(K))t^{d}=\prod_{j\geq
1}\left(\frac{1}{1-t^{j}}\right)^{\chi_{c}(\mathrm{Irr}_{j,n}(K))}.$

On the other hand, Lemma 3.2 and Lemma 3.3(4) show that

$\displaystyle\chi_{c}(\mathrm{Poly}_{d,n}(K))$
$\displaystyle=\chi_{c}(\mathbb{P}^{\binom{n+d}{n}-1}(K))-\chi_{c}(\mathbb{P}^{\binom{n+d-1}{n}-1}(K))$
$\displaystyle=\left[\binom{n+d}{n}\right]_{\chi_{c}(K)}-\left[\binom{n+d-1}{n}\right]_{\chi_{c}(K)}$
$\displaystyle=P_{d,n}(\chi_{c}(K)).$

The higher cyclotomic identity (Theorem 2.4) implies that

$\sum_{d\geq 0}P_{d,n}(\chi_{c}(K))t^{d}=\prod_{j\geq
1}\left(\frac{1}{1-t^{j}}\right)^{M_{j,n}(\chi_{c}(K))}.$

Hence by the uniqueness of Lemma 2.3 we conclude that for all $d,n\geq 1$,

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=M_{d,n}(\chi_{c}(\mathbb{R}))=M_{d,n}(-1),$

and similarly for $K=\mathbb{C}$. ∎

##  4\. Evaluating higher necklace polynomials at roots of unity

In this section we express the values of higher necklace polynomials
$M_{d,n}(\zeta_{p})$ at prime order roots of unity in terms of the balanced
base $p$ expansion of the number of variables $n$.

######  Definition 4.1.

Let $b\geq 2$ and $n\geq 1$ be integers. A balanced base $b$ expansion of $n$
is an expression

$n=b^{k_{2m}}-b^{k_{2m-1}}+b^{k_{2m-2}}-\ldots+b^{k_{2}}-b^{k_{1}},$

where $0\leq k_{0}<k_{1}<\ldots<k_{2m}$ is an increasing sequence of integers
of even length and the coefficients of the powers of $b$ alternate between
$\pm 1$.

######  Lemma 4.2.

Let $n\geq 0$ and $b\geq 2$ be integers. Then $n$ has a balanced base $b$
expansion if and only if the base $b$ expansion of $n$ only contains the
digits $0$ and $b-1$. Furthermore, if $n$ has a balanced base $b$ expansion,
then it is unique.

###### Proof.

First suppose that the base $b$ expansion of $n$ contains only the digits $0$
and $b-1$,

$n=(b-1)b^{k_{m}}+(b-1)b^{k_{m-1}}+\ldots+(b-1)b^{k_{1}}.$

Expanding each $(b-1)b^{k}=b^{k+1}-b^{k}$ term and collecting coefficients on
each power of $b$ gives a balanced base $b$ expansion of $n$. Note that each
nonzero base $b$ digit of $n$ contributes two terms to the balanced expansion
and cancellation occurs in pairs, hence the expansion we get in this way has
an even number of terms.

Next suppose that $n$ has a balanced base $b$ expansion

$n=b^{k_{2m}}-b^{k_{2m-1}}+b^{k_{2m-2}}-\ldots+b^{k_{2}}-b^{k_{1}}.$ (4.1)

If $k_{2\ell}\neq k_{2\ell-1}+1$, then we can replace $b^{2\ell}-b^{2\ell-1}$
in (4.1) with

$\displaystyle b^{k_{2\ell}}-b^{k_{2\ell-1}}$
$\displaystyle=b^{k_{2\ell}}-b^{k_{2\ell}-1}+b^{k_{2\ell}-2}-b^{k_{2\ell}-3}+\ldots+b^{k_{2\ell-1}+1}-b^{k_{2\ell-1}}$
$\displaystyle=(b-1)b^{k_{2\ell}-1}+(b-1)b^{2\ell-3}+\ldots+(b-1)b^{2\ell-1}.$

Note that here we are using that a balanced base $b$ expansion has an even
number of terms. Hence, after making this replacement for each consecutive
pair of terms in (4.1), we have a base $b$ expansion for $n$ which only
contains the digits $0$ and $b-1$.

The uniqueness of balanced base $b$ expansions then follows from the above
argument and the uniqueness of the usual base $b$ expansion of $n$. ∎

######  Example 4.3.

Every positive integer has a balanced base $2$ expansion since $0$ and $b-1=1$
are the only possible binary digits. For example the balanced base $2$
expansion of $n=55$ is

$55=2^{6}-2^{4}+2^{3}-1$

######  Theorem 4.4.

Let $p$ be a prime and let $n\geq 1$ be an integer such that

$n=\sum_{k\geq 0}b_{k}p^{k}$

is the balanced base $p$ expansion of $n$. If $\zeta_{p}$ is a primitive $p$th
root of unity, then

$M_{d,n}(\zeta_{p})=\begin{cases}b_{k}&\text{if }d=p^{k},\\\
0&\text{otherwise.}\end{cases}$

Thus it follows that $\Phi_{p}(x)$ divides $M_{d,n}(x)$ for all but finitely
many $d\geq 1$ whenever $n$ has a balanced base $p$ expansion.

Before proving Theorem 4.4 we prove two lemmas. Recall that if $m\geq 0$ is an
integer we write

$[m]_{x}:=\frac{x^{m}-1}{x-1}=x^{m-1}+x^{m-2}+\ldots+x+1.$

######  Lemma 4.5.

If $\zeta$ is a non-trivial $n$th root of unity, then $[m]_{\zeta}$ depends
only on $m$ modulo $n$.

###### Proof.

If $\zeta$ is a nontrivial $n$th root of unity, then

$[n]_{\zeta}=\zeta^{n-1}+\zeta^{n-2}+\ldots+\zeta+1=0.$

If $m=an+b$, then

$[m]_{x}=\frac{x^{an+b}-1}{x-1}=x^{b}\cdot\frac{x^{an}-1}{x^{n}-1}\cdot\frac{x^{n}-1}{x-1}+\frac{x^{b}-1}{x-1}=x^{b}[a]_{x^{n}}[n]_{x}+[b]_{x}.$

Evaluating at $x=\zeta$ gives

$[m]_{\zeta}=\zeta^{b}a[n]_{\zeta}+[b]_{\zeta}=[b]_{\zeta}.\qed$

Lemma 4.6 is a beautiful result due to Lucas [10]; see Fine [3] for a modern
proof.

######  Lemma 4.6 (Lucas’s Theorem).

If $p$ is a prime and

$\displaystyle m$
$\displaystyle=a_{k}p^{k}+a_{k-1}p^{k-1}+\ldots+a_{1}p+a_{0}$ $\displaystyle
n$ $\displaystyle=\hskip 1.0ptb_{k}p^{k}+\hskip
1.0ptb_{k-1}p^{k-1}+\ldots+b_{1}p\hskip 1.0pt+\hskip 1.0ptb_{0}$

are the base $p$ expansions of the natural numbers $m$ and $n$ (possibly with
leading zero coefficients), then

$\binom{m}{n}\equiv\binom{a_{k}}{b_{k}}\binom{a_{k-1}}{b_{k-1}}\cdots\binom{a_{1}}{b_{1}}\binom{a_{0}}{b_{0}}\bmod
p.$

We now prove Theorem 4.4.

###### Proof of Theorem 4.4.

Suppose that $n$ has a balanced base $p$ expansion and let $\zeta_{p}$ be a
non-trivial $p$th root of unity. Then by Theorem 2.4,

$\sum_{d\geq 0}P_{d,n}(\zeta_{p})t^{d}=\prod_{j\geq
1}\left(\frac{1}{1-t^{j}}\right)^{M_{j,n}(\zeta_{p})}.$ (4.2)

We evaluate $M_{d,n}(\zeta_{p})$ by expressing the left hand side of (4.2) as
an infinite product of the same form in another way and then appealing to the
uniqueness of Lemma 2.3. Towards that end, let
$Q(t)\in\Lambda(\mathbb{Q}(\zeta_{p}))$ be defined by

$Q(t):=\sum_{d\geq 0}\left[\binom{d+n}{n}\right]_{\zeta_{p}}t^{d}.$

Lemma 3.2 implies that $P_{d,n}(x)$ may be expressed as

$P_{d,n}(x)=\frac{x^{\binom{d+n}{n}}-x^{\binom{d+n-1}{n}}}{x-1}=\left[\binom{d+n}{n}\right]_{x}-\left[\binom{d+n-1}{n}\right]_{x}.$
(4.3)

Thus,

$\displaystyle\sum_{d\geq 0}P_{d,n}(\zeta_{p})t^{d}$
$\displaystyle=\sum_{d\geq
0}\left(\left[\binom{d+n}{n}\right]_{\zeta_{p}}-\left[\binom{d+n-1}{n}\right]_{\zeta_{p}}\right)t^{d}$
$\displaystyle=\sum_{d\geq
0}\left[\binom{d+n}{n}\right]_{\zeta_{p}}t^{d}-t\sum_{d\geq
1}\left[\binom{d+n-1}{n}\right]_{\zeta_{p}}t^{d-1}$ $\displaystyle=Q(t)-tQ(t)$
$\displaystyle=(1-t)Q(t).$

Next we determine the coefficients of $Q(t)$. Say positive integers $d$ and
$n$ are _$p$ -complementary_ if there is no $p^{k}$ with a non-zero
coefficient in the base $p$ expansions of both $d$ and $n$. Now suppose that
$n$ has a balanced base $p$ expansion. If $d$ and $n$ are not
$p$-complementary, suppose $p^{k}$ is the smallest power of $p$ common to the
base $p$ expansions of $d$ and $n$. Then the coefficient of $p^{k}$ in $d+n$
is 0 since

  1. (1)

the coefficient of $p^{k}$ in $n$ is $p-1$ by our assumption that $n$ has a
balanced base $p$ expansion,

  2. (2)

the coefficient of $p^{k}$ in $d$ is at least 1, and

  3. (3)

the minimality of $k$ implies there are no carries for smaller power $p$ in
the sum $d+n$.

Thus Lucas’s theorem (Lemma 4.6) implies that if $d$ and $n$ are not
$p$-complementary, then

$\binom{d+n}{n}\equiv 0\bmod p$

since the factor corresponding to $p^{k}$ will be 0. Therefore, if $d$ and $n$
are not $p$-complementary, then by Lemma 4.5 we have

$\left[\binom{d+n}{n}\right]_{\zeta_{p}}=[0]_{\zeta_{p}}=0.$

Next suppose $d$ and $n$ have base $p$ expansions

$\displaystyle d$
$\displaystyle=a_{k}p^{k}+a_{k-1}p^{k-1}+\ldots+a_{1}p+a_{0}$ $\displaystyle
n$ $\displaystyle=b_{k}p^{k}+b_{k-1}p^{k-1}+\ldots+b_{1}p+b_{0}.$

If $d$ and $n$ are $p$-complementary, then the base $p$ expansion of $d+n$ is

$d+n=(a_{k}+b_{k})p^{k}+(a_{k-1}+b_{k-1})p^{k-1}+\ldots+(a_{1}+b_{1})p+(a_{0}+b_{0})$

where for each $i$ at most one of $a_{i}$ and $b_{i}$ is non-zero. Lucas’s
theorem implies that

$\binom{d+n}{n}\equiv\binom{a_{k}+b_{k}}{b_{k}}\binom{a_{k-1}+b_{k-1}}{b_{k-1}}\cdots\binom{a_{1}+b_{1}}{b_{1}}\binom{a_{0}+b_{0}}{b_{0}}\bmod
p.$

Let $0\leq i\leq k$. If $a_{i}=0$, then

$\binom{a_{i}+b_{i}}{b_{i}}=\binom{b_{i}}{b_{i}}=1,$

and if $b_{i}=0$ then

$\binom{a_{i}+b_{i}}{b_{i}}=\binom{a_{i}}{0}=1.$

Therefore, if $d$ and $n$ are $p$-complementary, then

$\binom{d+n}{n}\equiv 1\bmod p.$

Hence by Lemma 4.5,

$\left[\binom{d+n}{n}\right]_{\zeta_{p}}=[1]_{\zeta_{p}}=1.$

Combining these computations we have

$Q(t)=\sum_{d\geq
0}\left[\binom{d+n}{n}\right]_{\zeta_{p}}t^{d}=\sum_{\begin{subarray}{c}d\text{
is $p$-comp.}\\\ \text{to }n\end{subarray}}t^{d}.$

The existence and uniqueness of base $p$ expansions of natural numbers is
equivalent to the following product formula,

$\frac{1}{1-t}=\sum_{d\geq 0}t^{d}=\prod_{k\geq
1}\sum_{a=0}^{p-1}t^{ap^{k}}=\prod_{k\geq
1}\frac{1-t^{p^{k+1}}}{1-t^{p^{k}}},$

where the factor of $\frac{1-t^{p^{k+1}}}{1-t^{p^{k}}}$ contributes to the
coefficient of $t^{d}$ precisely when $d$ is not $p$-complementary to $p^{k}$.
If $n=(p-1)p^{\ell_{1}}+(p-1)p^{\ell_{2}}+\ldots+(p-1)p^{\ell_{s}}$ is the
base $p$ expansion of $n$ (which can be expressed in this form by the
assumption that $n$ has a balanced base $p$ expansion,) then

$Q(t)=\sum_{\begin{subarray}{c}d\text{ is $p$-comp.}\\\ \text{to
}n\end{subarray}}t^{d}=\frac{1}{1-t}\prod_{i=1}^{s}\frac{1-t^{p^{\ell_{i}}}}{1-t^{p^{\ell_{i}+1}}}.$

Therefore

$\sum_{d\geq
0}P_{d,n}(\zeta_{p})t^{d}=(1-t)Q(t)=\prod_{i=1}^{s}\frac{1-t^{p^{\ell_{i}}}}{1-t^{p^{\ell_{i}+1}}}=\prod_{k\geq
1}\left(\frac{1}{1-t^{p^{k}}}\right)^{b_{k}},$

where $n=\sum_{k\geq 0}b_{k}p^{k}$ is the balanced base $p$ expansion of $n$.
The uniqueness of product expansions of this form provided by Lemma 2.3
implies that $M_{p^{k},n}(\zeta_{p})=b_{k}$ and $M_{d,n}(\zeta_{p})=0$ when
$d$ is not a power of $p$. ∎

For a fixed $n$ there are finitely many primes $p$ for which $n$ has a
balanced base $p$ expansion. Theorem 4.4 tells us that for each such prime $p$
there are only finitely many $d$ such that $M_{d,n}(\zeta_{p})\neq 0$ for
$\zeta_{p}$ a primitive $p$th root of unity.

######  Remark.

When $n$ does not have a balanced base $p$ expansion, one can still use the
methods of Theorem 4.4 to determine the values of $M_{d,n}(\zeta_{p})$, but
the results become more challenging to state concisely. Since our main
interest is in the case $p=2$ and its application to evaluating Euler
characteristics, we defer this more refined analysis.

We end this section with an evaluation of $M_{d,n}(1)$ which will be used
below to compute the compactly supported Euler characteristics of
$\mathrm{Irr}_{d,n}(\mathbb{C})$.

######  Proposition 4.7.

For all $d,n\geq 1$,

$M_{d,n}(1)=\begin{cases}n&\text{if }d=1,\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

Note that $[m]_{1}=m$, hence by (4.3) we have

$P_{d,n}(1)=\binom{d+n}{n}-\binom{d+n-1}{n}=\binom{d+n-1}{d}=\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{n}{d}\right)\kern-3.00003pt\right).$

Therefore

$\sum_{d\geq 0}P_{d,n}(1)t^{d}=\sum_{d\geq
0}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{n}{d}\right)\kern-3.00003pt\right)t^{d}=\left(\frac{1}{1-t}\right)^{n}.$

Thus the higher cyclotomic identity and the uniqueness of Lemma 2.3 imply that
$M_{1,n}(1)=n$ and $M_{d,n}(1)=0$ for $d>1$. ∎

##  5\. Conclusion

Combining the results of the previous sections we now arrive at the main
result.

######  Theorem 5.1.

Let $d,n\geq 1$ and suppose that $n=\sum_{k=0}^{\ell}b_{k}2^{k}$ is the
balanced binary expansion of $n$. Then

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=\begin{cases}b_{k}&\text{if
}d=2^{k},\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

Theorem 3.4 implies that
$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{R}))=M_{d,n}(-1)$ and Theorem 4.4
expresses the value of $M_{d,n}(-1)$ in terms of the balanced binary expansion
of $n$. ∎

Similarly Proposition 4.7 gives the following computation of the Euler
characteristic of $\mathrm{Irr}_{d,n}(\mathbb{C})$.

######  Proposition 5.2.

Let $d,n\geq 1$. Then

$\chi_{c}(\mathrm{Irr}_{d,n}(\mathbb{C}))=\begin{cases}n&\text{if }d=1,\\\
0&\text{otherwise.}\end{cases}$

As noted in the introduction, Theorem 5.1 suggests that the space
$\mathrm{Irr}_{d,n}(\mathbb{R})$ may have an interesting cell decomposition
determined by the binary expansion of the number of variables $n$.

## References

  * [1] W. Chen, Stability in the cohomology of the space of complex irreducible polynomials in several variables, _Int. Math. Res. Not._ , (2019).
  * [2] J. Elliott, Binomial rings, integer-valued polynomials, and $\lambda$-rings, _J. Pure Appl. Algebra_ , 207, no. 1, (2006), 165-185, DOI: 10.1016/j.jpaa.2005.09.003.
  * [3] N. J. Fine, Binomial coefficients modulo a prime, _Am. Math. Monthly_ , 54, no. 10, (1947), 589-592, DOI: 10.2307/2304500.
  * [4] P. Hall, The Edmonton Notes on Nilpotent Groups, _Queen Mary College Mathematics Notes_ , Mathematics Department, Queen Mary College, London, (1969).
  * [5] A. Hatcher, Algebraic topology, _Cambridge University Press_ , (2002), available at
http://pi.math.cornell.edu/ hatcher/AT/AT.pdf.

  * [6] T. Hyde, Liminal reciprocity and factorization statistics, _Alg. Comb._ , 2, no. 4, (2019), 521-539.
  * [7] T. Hyde, Cyclotomic factors of necklace polynomials, _preprint_ , (2018), arXiv: 1811.08601.
  * [8] T. Hyde, Polynomial statistics, necklace polynomials, and the arithmetic dynamical Mordell-Lang conjecture, _dissertation_ , University of Michigan, (2019).
  * [9] T. Hyde, A. Odesky, Cyclotomic factors of necklace polynomials and unlikely evaluations, _in preparation_ , (2020).
  * [10] E. Lucas, Théorie des fonctions numériques simplement périodiques, _Am. J. Math._ , 1, no. 4, (1878), 289-321, DOI: 10.2307/2369373.
  * [11] I. G. MacDonald, The Poincaré polynomial of a symmetric product, _Math. Proc. Camb. Philos. Soc._ , 58, no. 4, (1962), DOI: 10.1017/S0305004100040573.
  * [12] E. H. Spanier, Algebraic Topology, Springer Science & Business Media, (1989).
  * [13] R. Vakil, Arizona winter school notes, available at http://swc.math.arizona.edu/aws/2015/2015VakilNotes.pdf.