\documentclass{amsart}
\def\demi{\textstyle{\frac{1}{2}}}
\def\eps{\varepsilon }
\def\gam{\gamma}
\def\Bbb{\mathbb }
\def\RR{\mathbb R}
\def\CC{\mathbb C}
\def\RRR{\overline{\mathbb R}}
\def\uu{\mathcal U}
\newcommand{\set}[1]{\{#1\}}%set

\newtheorem{theorem}{Theorem}%[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem*{definition}{Definition}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{remarks}{Remarks}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{problems}[theorem]{Problems}

%\numberwithin{equation}{section}

\begin{document}
\title{The smallest univoque number is not isolated}
\author{Vilmos Komornik}
\address{Institut de Recherche Math\'ematique Avanc\'ee\\
         Universit\'e Louis Pasteur et CNRS\\
         7, rue Ren\'e Descartes\\
         67084 Strasbourg Cedex, France}
\email{komornik@math.u-strasbg.fr}
\author{Paola Loreti}
\address{Dipartimento di Metodi e Modelli\\
          Matematici per le Scienze Applicate\\
          Universit\`a di Roma ``La Sapienza''\\
          Via A. Scarpa, 16\\
          00161 Roma, Italy}
\email{loreti@dmmm.uniroma1.it}
\author{Attila Peth\H o}
\address{Department of Computer Science\\
University of Debrecen\\
 H-4010 Debrecen\\
P.O. Box 12, Hungary}
\email{pethoe@neumann.math.klte.hu}
\thanks{{\it Mathematics Subject Classification:} 11A63, 11A67, 11B85\\
{\it Key words and phrases:} $\beta$-expansion, univoque number,
Thue-Morse sequence.\\
 The research of the first two authors was
partially supported by the Consiglio Nazionale delle Ricerche. The
research of the third author was partially supported by Hungarian
National Foundation for Scientific Research, Grant No. T29330 and
38225.}
%\author{}
%\address{}
%\curraddr{}
%\email{}
%\subjclass{}
%\keywords{}
\date{\today}
%\thanks
\dedicatory{Dedicated to the 80th birthday of Professor Lajos
Tam\'assy.}




\begin{abstract}
Komornik and Loreti \cite{KomLor95} showed that there exists a
smallest univoque number $q' \approx 1.787$. Later Allouche and
Cosnard \cite{AllCos2000} proved that this number is
transcendental. The aim of this note is to construct a
(decreasing) sequence of algebraic numbers converging to $q'$.
\end{abstract}
\maketitle


\section{Introduction}\label{s1}

Given a real number $1\le q\le 2$, there exists at least one
sequence $(c_i)$ of zeroes  and ones satisfying the equality
\begin{equation}\label{11}
1=\frac{c_1}{q}+\frac{c_2}{q^2}+\frac{c_3}{q^3}+\dots
\end{equation}
One such sequence, denoted by $(\gam_i)$, can be obtained by the
so-called {\em greedy} algorithm of R\'enyi \cite{Ren1957}:
proceeding by induction, we choose $c_i=1$ whenever possible.
Among all expansions for a given $q$, this is lexicographically
the largest.

If $q=2$, then this is the unique possible expansion: $c_i=1$ for all $i$. Erd\H os,
Horv\'ath and Jo\' o \cite{ErdHorJoo1991} discovered that there exist also smaller numbers
$q$ having this curious uniqueness property; following Dar\'oczy and K\'atai
\cite{DarKat1993} we call them {\em univoque} numbers. Subsequently, they were
characterized  algebraically in \cite{ErdJooKom55} (see also \cite{KomLor116}
for an extension of this result):

\begin{theorem}\label{t1}
A number $1\le q\le 2$ is univoque if and only if there exists an expansion $(\gam_i)$ of
$1$ satisfying the following two conditions (in the lexicographic sense):
\begin{equation}\label{12}
\gam_{i+1}\gam_{i+2}\dots <\gam_1\gam_2\dots\quad\text{whenever}\quad \gam_i=0
\end{equation}
and
\begin{equation}\label{13}
\overline{\gam_{i+1}\gam_{i+2}\dots} <\gam_1\gam_2\dots\quad\text{whenever}\quad
\gam_i=1.
\end{equation}
\end{theorem}
Here and in the sequel we use the notation $\overline{c}:=1-c$.

Among several interesting properties of the set $\uu$ of univoque numbers, for which we
refer to the papers  \cite{AllCos2000}, \cite{AllCos2001}, \cite{DarKat1993}, \cite{DarKat1995},
\cite{ErdHorJoo1991},
\cite{Kal1999} and \cite{KomLor95},
we recall from \cite{KomLor95} that
there exists a smallest univoque number $q'\approx 1.787$, and the
corresponding expansion is given by the truncated Thue--Morse sequence
\begin{equation*}
(\tau_i)_{i=1}^\infty= 1101\ 0011\dots
\end{equation*}

The purpose of this note is to investigate the following two questions: \mbox{}

\begin{itemize}
\item One may wonder whether $q'$ is an isolated univoque number or not. In the first case
one could look for the second smallest univoque number, and so on.

\item Allouche and Cosnard proved in \cite{AllCos2000} that $q'$ is transcendental. It is
than natural to look for the smallest {\em algebraic} univoque number if it exists.
\end{itemize}

Both problems are solved by the following

\begin{theorem}\label{t2}
There exists a (decreasing) sequence of algebraic univoque numbers
converging to $q'$. In particular, $q'$ is not an isolated point
of $\uu$.
\end{theorem}

\section{Proof of theorem \ref{t2}}\label{s2}

For the purpose of the present paper, it is advantageous to adopt
the following definition of the Thue--Morse sequence $(\tau_i)$:
if
\begin{equation*}
i=\eps_k2^k+\dots+\eps_0
\end{equation*}
is the dyadic expansion of some nonnegative integer $i$, then we define
\begin{equation}\label{1}
\tau_i:=
\begin{cases}
1&\text{ if $\eps_k+\dots+\eps_0$ is odd,}\\
0&\text{ if $\eps_k+\dots+\eps_0$ is even.}
\end{cases}
\end{equation}
In particular, $\tau_0=0$. See \cite{KomLor95} for its
equivalence with another usual definition.

Our main tool is the following strenghtening of a property of the
Thue--Morse sequence $\tau_1$, $\tau_2$,\dots, established in  \cite{KomLor95}.

\begin{lemma}\label{l3}
Let $1\le i<2^{N+1}$ for some nonnegative integer $N$. \mbox{}\smallskip

(a) If $\tau_i=0$, then $\tau_{i+1}\dots\tau_{i+2^N}<\tau_{1}\dots\tau_{2^N}$ in the lexicographic
sense.\smallskip

(b) If $\tau_i=1$, then $\overline{\tau_{i+1}\dots\tau_{i+2^N}}<\tau_{1}\dots\tau_{2^N}$ in the lexicographic
sense.
\end{lemma}

\begin{remark}
In fact, part (a) remains valid even if $\tau_i=1$, except  the case where $N=0$ and $i=1$,
while part (b) remains always valid even if $\tau_i=0$. An analogous property was established
recently by Glendinning and Sidorov  \cite{GleSid2000}.
\end{remark}

\begin{proof}
Consider first the case $\tau_i=0$. Then $\eps_k+\dots+\eps_0$ is even and therefore $\eps_k+\dots+\eps_0\ge 2$
because $i\ge
1$ by assumption. Hence we may write $i=2^n+2^m+j$ with $2^n>2^m>j\ge 0$. We claim that
\begin{equation}\label{2}
\tau_{i+1}\dots\tau_{i+2^N}<\tau_{j+1}\dots\tau_{j+2^N}.
\end{equation}

We distinguish two cases. If $n\ge m+2$, then using \eqref{1} we have
\begin{equation*}
\tau_{i+k}=\tau_{j+k}\quad\text{for}\quad 1\le k<2^m-j
\end{equation*}
but
\begin{equation*}
\tau_{i+2^m-j}=\tau_{2^n+2^{m+1}}=0<1=\tau_{2^m}=\tau_{j+2^m-j}.
\end{equation*}
Since
\begin{equation*}
2^m-j\le 2^m\le 2^{N-1}<2^N,
\end{equation*}
this proves \eqref{2}.

If $n=m+1$, then using \eqref{1} we obtain by a similar reasoning that
\begin{equation*}
\tau_{i+k}=\tau_{j+k}\quad\text{for}\quad 1\le k<2^{m+1}-j
\end{equation*}
but
\begin{equation*}
\tau_{i+2^{m+1}-j}=\tau_{2^{m+2}+2^{m}}=0<1=\tau_{2^{m+1}}=\tau_{j+2^{m+1}-j}.
\end{equation*}
Since
\begin{equation*}
2^{m+1}-j\le 2^{m+1}=2^n\le 2^N,
\end{equation*}
\eqref{2} follows again.

Since $\tau_j=\tau_i=0$, we may iterate \eqref{2} until we obtain $j=0$, thereby proving the desired
inequality.\medskip

Now consider the case $\tau_i=1$ and write $i=2^m+j$ with $2^m>j\ge 0$. We claim that
\begin{equation}\label{3}
\overline{\tau_{i+1}\dots\tau_{i+2^N}}<\tau_{j+1}\dots\tau_{j+2^N}.
\end{equation}
Indeed, using \eqref{1} we have
\begin{equation*}
\overline{\tau_{i+k}}=\tau_{j+k}\quad\text{for}\quad 1\le k<2^{m}-j
\end{equation*}
but
\begin{equation*}
\overline{\tau_{i+2^{m}-j}}=\overline{\tau_{2^{m+1}}}=0<1=\tau_{2^m}=\tau_{j+2^{m}-j}.
\end{equation*}
Since
\begin{equation*}
2^{m}-j\le 2^{m}\le 2^N,
\end{equation*}
this proves \eqref{3}.

If $j=0$, then we are done. If $j>0$, then we complete the proof by combining \eqref{2} and \eqref{3}.
\end{proof}

Now fix a nonnegative integer $N$ and introduce the following sequence:
\begin{equation}\label{4}
c_i:=
\begin{cases}
\tau_i&\text{ if $1\le i<2^{N+1}$,}\\
c_{i-2^N}&\text{ if $i\ge 2^{N+1}$.}
\end{cases}
\end{equation}
This sequence was used for different purposes in
a recent work of Glendinning and Sidorov  \cite {GleSid2000}.
Observe that the sequence $(c_n)$ is periodic with period $2^N$ beginning with $c_{2^N}$. Let us write down the
first 16
elements of the Thue--Morse sequence and of the sequences $(c_n)$ for $N=0, 1, 2$:

\begin{align*}
&(\tau_i):\quad &1101\ 0011\ 0010\ 1101\dots\\
&N=0:           &1111\ 1111\ 1111\ 1111\dots\\
&N=1:           &1101\ 0101\ 0101\ 0101\dots\\
&N=2:           &1101\ 0011\ 0011\ 0011\dots
\end{align*}

Let us note for further reference that

\begin{equation}\label{5}
\tau_i=\tau_{i-2^N}\quad\text{for}\quad 2^{N+1}\le i<2^{N+1}+2^N.
\end{equation}
Indeed, this follows easily from \eqref{1}.

It is clear that the equation
\begin{equation}\label{6}
1=\frac{c_1}{q}+\frac{c_2}{q^2}+\frac{c_3}{q^3}+\dots
\end{equation}
defines an algebraic number $1<q_N\le 2$  satisfying $q_N\to q'$ as $N\to\infty$.

\begin{proof}[Proof of Theorem \ref{t2}]
Thanks to Theorem \ref{t1}, it suffices to verify that the
sequence  $(c_n)$ is admissible in the following sense:
\begin{equation}\label{7}  c_{i+1}\dots c_{i+2^N}<c_{1}\dots
c_{2^N}\quad\text{whenever}\quad c_i=0 \end{equation}
and
\begin{equation}\label{8}
\overline{c_{i+1}\dots c_{i+2^N}}<c_{1}\dots c_{2^N}\quad\text{whenever}\quad c_i=1.
\end{equation}

For $1\le i<2^{N+1}$ both relations follow from the similar
properties of the Thue--Morse sequence established in the
preceding lemma because the first $2^{N+1}+2^{N}-1$ of the two
sequences coincide by equation \eqref{5}.

For $i\ge  2^{N+1}$ the relations \eqref{7} and \eqref{8} now follow by induction because the sequences
$c_{i+1}\dots
c_{i+2^N}$ and $c_{i+1-2^N}\dots c_{i}$ coincide, and also $c_i=c_{i-2^N}$, so that $c_i=0$ implies
$c_{i-2^N}=0$ and $c_i=1$
implies $c_{i-2^N}=1$.
\end{proof}

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\end{thebibliography}



\end{document}

JFM 37.0066.17
Thue, A.
�ber unendliche Zeichenreihen. [B] Christiana Vidensk. Selsk. Skr. 1906. Nr. 7. 22 S. Lex. $8^{\circ}$.
Published: (1906)

JFM 44.0462.01
Thue, A.
�ber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. (German)
[J] Christiania Vidensk. Selsk. Skr. 1912, 1, No. 1, 67 S. (1912).
Published: 1912
