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\begin{document}

\begin{center}

 {\Large Results and problems on diophantine properties of radix representations}\\
 \vspace{1.5cm}

{\large  Attila Peth\H{o} }

{Department of Computer Science\\ University of Debrecen, 
Debrecen, Hungary} \vspace{1.5cm}

Representation Theory XVI, Number Theory Section\\
IUC, Dubrovnik, Croatia, June 26, 2019. \vspace{1cm}

{\small Talk is based partially on joint works with Jan-Hendrik Evertse, K\'alm\'an Gy\H{o}ry and J\"org Thuswaldner.}


\end{center}


\newpage

{\bf 1. Radix representation in number fields}

Let $g,h\ge 2$. Denote $(n)_g$ the sequence of digits of the $g$-ary representation of $n$, e.g. $(2018)_{10}=2018, (2018)_5=31033$.\\[0.3ex]
Let $\K$ an algebraic number field with ring of integers $\Z_{\K}$.\\[0.3ex]
$\LL$ a finite extension of $\K$ with ring of integers $\Z_{\LL}$.\\[0.3ex]
The pair $(\gamma,\DD)$, where $\gamma\in \Z_{\LL}$ and $\DD$ is a complete residue system modulo $\gamma$, in $\Z_{\K}$ is called a GNS in $\Z_\LL$ if for any $0\not=\beta\in \Z_{\LL}$ there exist an integer $\ell\ge 0$ and $a_0,\dots,a_{\ell}\in \D, a_{\ell}\not= 0$ such that
\begin{equation}\label{ketto}
  \beta = a_{\ell} \gamma^{\ell}+\dots + a_1 \gamma + a_0 .
\end{equation}

Denote the sequence or word of the digits $a_{\ell}\dots a_1a_0$ by $(\beta)_{\gamma}$.

The GNS concept was initiated by D. Knuth, and developed further by Penney, I. K\'atai, J. Szab\'o, B. Kov\'acs, etc.

Not all $(\gamma,\DD)$ is a GNS! For example $\left(\frac{-1+\sqrt{-7}}{2}, \{0.1\}\right)$ is, but $\left(\frac{1+\sqrt{-7}}{2}, \{0.1\}\right)$ is not a GNS in $\Z[\sqrt{-7}]$.

This GNS is a special case of GNS in polynomial ring over an order, i.e., a commutative ring with unity, whose additive structure is a free $\Z$-module of finite rank. To avoid technical difficulties we restrict ourself to maximal orders of number fields. The GNS property is decidable in the general setting.\\[1ex]

\begin{problem}
  Let $\DD\subset \Z_{\K}$ be given. How many $\gamma \in \Z_{\LL}$ exist such that $(\gamma,\DD)$ is a GNS in $\Z_\LL$?
\end{problem}

For $\K=\Q$ the answer is: at most one! If $\DD\subset \Z \subset \Z_{\K}$ then there are only finitely many, effectively computable. (Idea of the proof later.) In general the problem is open.

\newpage


{\bf 2. A theme of K. Mahler}

K. Mahler, 1981, proved that the number $0.(1)_{g}(h)_{g}(h^2)_{g} \ldots$ is irrational, equivalently: the infinite word $(1)_{g}(h)_{g}(h^2)_{g} \ldots$ is not periodic. Refinements, generalizations and new methods by\\
$\bullet$ P. Bundschuh, 1984\\
$\bullet$ H. Niederreiter, 1986\\
$\bullet$ Z. Shan, 1987\\
$\bullet$ Z. Shan and E. Wang, 1989: Let $(n_i)_{i=1}^{\infty}$ be a strictly increasing sequence of integers. Then $0.(g^{n_1})_h(g^{n_2})_h\ldots$ is irrational. In the proof they used the theory of Thue equations.\\

\newpage
Generalizations for numeration systems based on linear recursive sequences:

$\bullet$ P.G. Becker, 1991\\
$\bullet$ P.G. Becker and J. Sander 1995\\
$\bullet$ G. Barat, R. Tichy and R. Tijdeman, 1997\\
$\bullet$ G. Barat, C. Frougny and A. Peth\H{o}, 2005

\newpage
{\bf 3.1. Results on power sums}

Let $0\notin \mathcal{A}, \mathcal{B}\subset \Z_{\LL}$ be finite, and $\Gamma, \Gamma^+$ be the semigroup, group generated by $\mathcal{B}$. Put $$S(\mathcal{A},\mathcal{B},s)= \{\alpha_{1}\mu_{1}+\dots+\alpha_{s}\mu_{s}\; :\; \alpha_{j} \in \mathcal{A}, \mu_{j}\in \Gamma\}.$$

Example: $\LL=\Q, \mathcal{A}=\{1\}, \mathcal{B}=\{2,3\}$ then $$S(\mathcal{A},\mathcal{B},2)=\{2^{a}3^b + 2^c3^d\;:\; a,b,c,d\ge 0\}.$$

\newpage
\begin{theorem}\label{Mcor2}
  Let $s\ge 1$ and $\mathcal{A}, \mathcal{B}$ as above. Let $(c_n)$ be such that $c_n\in S(\mathcal{A},\mathcal{B},s)$. If $(\gamma,\DD)$ is a GNS in $\Z_{\LL}$, $\gamma\notin \Gamma^+$ and $(c_n)$ has infinitely many distinct terms then the infinite word $(c_1)_{\gamma}(c_2)_{\gamma}\ldots$ is not periodic.
\end{theorem}

Let $(c_1)_{\gamma}(c_2)_{\gamma}\ldots=f_0f_1\ldots$. Then
$$
g= \sum_{j=0}^{\infty} f_j \gamma^{-j}
$$
is a well defined complex number. A result of B. Kov\'acs and I. K\"ornyei, 1992 implies $g\notin \Q$. We expect at least $g\notin \LL$, but we are unable to prove this.

The proof of Theorem 1 is based on the following\\[1ex]

\begin{Le}
  For any $w\in \DD^*$ there are only finitely many $U\in  S(\mathcal{A},\mathcal{B},s)$ such that $(U)_{\gamma}=w_1w^k$, where $w_1$ is a suffix of $w$.
\end{Le}

\begin{proof}
  Let $w=d_0\ldots d_{h-1}$. If $(U)_{\gamma}=w_1w^k$ then $w_1=\lambda$ or $w_1=d_t\ldots d_{h-1}$. Set $q_0=0$ if $w_1=\lambda$, and $q_0 = d_t +d_{t+1}\gamma + \ldots + d_{h-1} \gamma^{h-t-1}$ otherwise. Further let $q = d_0 +d_{1}\gamma + \ldots + d_{h-1} \gamma^{h-1}$. We also have $U=\alpha_{1}\mu_{1}+\dots+\alpha_{s}\mu_{s}$. Then
  \begin{eqnarray*}
  % \nonumber % Remove numbering (before each equation)
    \alpha_{1}\mu_{1}+\dots+\alpha_{s}\mu_{s} &=& q_0 + \gamma^{h-t}\sum_{i=0}^{k-1} q\gamma^{ih} \\
     &=& q_0 + q  \gamma^{h-t}\frac{\gamma^{hk}-1}{\gamma^h -1} \\
     &=& \frac{q\gamma^{h-t}}{\gamma^h-1} \gamma^{hk} + q_0 -\frac{q \gamma^{h-t}}{\gamma^{h}-1}.
  \end{eqnarray*}

Setting
%\begin{eqnarray*}
$$
\alpha_{s+1}= \frac{q\gamma^{h-t}}{\gamma^h-1}, \qquad
\alpha_{s+2}= q_0 -\frac{q \gamma^{h-t}}{\gamma^{h}-1}
$$
%\end{eqnarray*}

we get the equation

\begin{equation}\label{Sunitspec}
  \alpha_{1}\mu_{1}+\dots+\alpha_{s}\mu_{s} = \alpha_{s+1} \gamma^{hk} + \alpha_{s+2}.
\end{equation}

As $(\gamma,\DD)$ is a GNS $|\gamma|>1$, hence $\gamma^h\not=1$ and $\alpha_{s+1},\alpha_{s+2}$ are well defined. Plainly $\alpha_{j}\in \LL, j=1,\ldots,s+2$ and $\alpha_{j}\not=0, k=1,\ldots,s$ by assumption. It is easy to see that $\alpha_{s+1}\not=0$ holds too.

\newpage
Taking $\Gamma_1$ the multiplicative semigroup generated by $\gamma$ and $b\in \mathcal{B}$  \eqref{Sunitspec} is a $\Gamma_1$-unit equation. If there are infinitely many $U\in  S(\mathcal{A},\mathcal{B},s)$ such that $(U)_{\gamma}=w_1w^k$ then $k$ can take arbitrary large values and \eqref{Sunitspec} has infinitely many solutions in $(\mu_1,\dots,\mu_s,\gamma^{hk})\in \Gamma_1^{s+1}$. By the theory of weighted $S$-unit equations the assumption $\gamma\notin \Gamma^+$ excluded this.
\end{proof}

\newpage
{\it Proof of Theorem \ref{Mcor2}.} Let $W=(c_1)_{\gamma}(c_2)_{\gamma}\ldots$ and assume that it is eventually periodic. Omitting, if necessary, some starting members of $(c_n)$ we may assume that it is periodic, i.e. $W=H^{\infty}$ with $H\in \DD^h$.

 There exist for all $n\ge 1$ a suffix $c_{n0}$ a prefix $c_{n1}$ of $H$ and an integer $e_n\ge 0$ such that $(c_n)_{\gamma}=c_{n0}H^{e_n}c_{n1}$.

  There exist only finitely many, elements of $\Z_{\K}$ with a $(\gamma,\DD)$-representation of bounded length. Thus, the length of the words $(c_n)_{\gamma}, n=1,2,\ldots$ is not bounded. Further, there are only $|\mathcal{A}|^s$ possible choices for the $s$-tuple $(a_{n1},\dots,a_{ns})$. Thus, there exists an infinite sequence $k_1<k_2<\ldots$ of integers such that $l((c_{k_n})_{\gamma})\ge h$ and $l((c_{k_{n+1}})_{\gamma})> l((c_{k_n})_{\gamma})$ and the $s$-tuples $(a_{k_n1},\dots,a_{k_ns})$ are the same for all $n\ge 1$.

  Write $(c_{k_n})_{\gamma} = c_{k_n0}H^{e_{k_n}}c_{k_n1}$, where $c_{k_n0}$ is a suffix and $c_{k_n1}$ is a prefix of $H$ for all $n\ge 1$. As $H$ has at most $h-1$ proper prefixes and $h-1$ proper suffixes there exists an infinite subsequence of $k_n, n\ge 1$ such that the corresponding words satisfy $c_{k_n0}=C_0$ and $c_{k_n1}=C_1$. In the sequel we work only with this subsequence, therefore we omit the subindexes.

  With this simplified notation we have $(c_n)_{\gamma}=C_0H^{e_n}C_1$, where $C_0$ denotes a proper suffix, and $C_1$ a proper  prefix of $H$ and $(e_n)$ tends to infinity. Finally, replacing $H$ by the suffix of length $h$ of $HC_1$, and denoting it again by $H$ we have $(c_n)_{\gamma}=C_0H^{e_n}$. This contradicts Lemma 1. $\Box$

\newpage
Considering for $\K=\Q$ the ordinary $g$-ary representation of integers we get immediately the following far reaching generalization of Mahler's result.\\[1ex]

\begin{Cor} \label{Minteger}
  Let $\mathcal{A}, \mathcal{B}$ be finite sets of positive integers and $g\ge 2$ be a positive integer. Let $\Gamma=\Gamma(\mathcal{B})$ and $c_n=a_{n1}u_{n1}+\dots+a_{ns}u_{ns}$ with $u_{ni}\in \Gamma, a_{ni}\in \mathcal{A}, 1\le i\le s, n\ge 1$. If $g\notin \Gamma$ and $(c_n)$ is not bounded, then $0.(c_1)_g(c_2)_g...$ is irrational.
\end{Cor}

\newpage
To illustrate the power of Theorem \ref{Mcor2} we formulate a further corollary.\\[1ex]

\begin{Cor} \label{Mcor3}
  Let $\gamma$ be an algebraic integer, which is neither rational nor imaginary quadratic. Let $\K=\Q(\gamma)$, $\DD$ be a complete residue system modulo $\gamma$ in $\Z_{\K}$ and $(\gamma,\DD)$ be a GNS in $\Z[\gamma]$. If $(c_n)$ is a sequence of elements of $\Z[\gamma]$ of given norm, which includes infinitely many pairwise different terms, then the word $(c_1)_{\gamma}(c_2)_{\gamma}\ldots$ is not periodic.
\end{Cor}

\begin{proof}
  There exists in $\Z_{\K}$ only finitely many pairwise not associated elements with given norm. Let $\mathcal{A}$ be such a set. There exist by Dirichlet's theorem $\varepsilon_1,\dots, \varepsilon_r$ such that every unit of infinite order of $\Z_{\K}$ can be written in the form $\varepsilon_1^{m_1}\cdots \varepsilon_r^{m_r}$. Setting $\mathcal{B}=\{\varepsilon_1,\dots, \varepsilon_r\}$ apply Theorem \ref{Mcor2}.
\end{proof}

Notice that in the rational and in the imaginary quadratic fields there are only finitely many elements with given norm, hence there are cases, when $(c_1)_{\gamma}(c_2)_{\gamma}\ldots$ is, and other cases, when it is not periodic.\\

\begin{problem}
  Let $A>0$ and $B\ge \max\{2,A\}$. Establish all repunits with respect to the GNS $\left(\frac{-A+\sqrt{A^2-4B}}{2}, \{0,1,\ldots,B-1\}\right)$ for various values of $A,B$.
\end{problem}
\newpage

{\bf 3.2. Results on rational integers}

We consider analogous questions on rational integers.\\[0.5ex]

\begin{theorem}\label{Mahlerint}
  Let $[\LL:\Q]=\ell\ge 2$ and $\gamma\in \Z_{\LL}, \DD\subset \Z$ such that $\gamma^{\ell}\notin \Z$ and $\DD$ is a complete residue system modulo $\gamma$. Assume that $(\gamma, \DD)$ is a GNS in $\Z_{\LL}$. Let $n_1,n_2,\ldots$ be an unbounded sequence of rational integers. Then $0.(n_1)_{\gamma}(n_2)_{\gamma}\ldots \notin \Q$.
\end{theorem}

Similarly to Theorem \ref{Mcor2} the proof is rooted in\\[0.5ex]

\begin{Le}\label{perword}
  Let $[\LL:\Q]=\ell\ge 2$ and $\gamma\in \Z_{\LL}, \DD\subset \Z$ such that $\gamma^{\ell}\notin \Z$ and $\DD$ is a complete residue system modulo $\gamma$. Assume that $(\gamma, \DD)$ is a GNS in $\Z_{\LL}$. For any $w\in \DD^*$ there are only finitely many $n\in \Z$ such that $(n)_{\gamma} = w_1w^k$, where $w_1$ is a suffix of $w$. 
\end{Le}

A simple consequence of this lemma is\\[1ex]

\begin{Cor}\label{repunit}
  Let $\LL,\gamma, \DD$ be as in Lemma \ref{perword}. There are only finitely many rational integers, which are repunits in the GNS $(\gamma,\DD)$, i.e., $(n)_{\gamma}=1^k$.
\end{Cor}

{\bf Scats of the proof of Corollary \ref{repunit}.} We have $\Q(\gamma)=\LL$, thus the degree of $\gamma$ is $\ell$. Denote $\gamma^{(j)}, j=1,\ldots,\ell$ the conjugates of $\gamma$. We have:\\[0.5ex]
$\bullet$ $|\gamma^{(j)}|>1, j=1,\ldots,\ell$ because $(\gamma,\DD)$ is a GNS.\\[0.5ex]
$\bullet$ If $1\le i<j\le \ell$ then $\gamma^{(i)}$ and $\gamma^{(j)}$ are multiplicatively independent by Dobrowolski, 1979.\\[1ex]
  
  If $n\in \Z$ such that $(n)_{\gamma}=1^k$ with some $k$ then $n =\sum_{j=0}^{k-1}\gamma^j=\frac{\gamma^k -1}{\gamma-1}$. Let $\gamma'\not=\gamma$ be a conjugate of $\gamma$. We may assume $1<|\gamma'|\le|\gamma|$, but $\gamma'/\gamma$ is not a root of unity. Then $n =\sum_{j=0}^{k-1}\gamma'^j=\frac{\gamma'^k -1}{\gamma'-1}$ too. Thus
  $$
  \frac{\gamma^k -1}{\gamma-1} = \frac{\gamma'^k -1}{\gamma'-1}
  $$
  or, equivalently,
  $$
  \left(\frac{\gamma}{\gamma'}\right)^k -1 = \frac{\gamma'-\gamma}{\gamma-1} \frac{1}{\gamma'^k}.
  $$
  If $|\gamma'|< |\gamma|$ simply analysis, otherwise Bakery. $\Box$
  
  \newpage

{\bf 3.3. Solutions of norm form equations}

Let $\K$ be an algebraic number field of degree $k$. It has $k$ isomorphic images, $\K^{(1)}=\K,\dots, \K^{(k)}$ in $\C$. Let $\alpha_1=1,\alpha_2,\dots,\alpha_s\in \Z_{\K}$ be $\Q$-linear independent elements and $L({\bf X})=\alpha_1X_1+\dots+ \alpha_sX_s$. Plainly $s\le k$. Consider the norm form equation
\begin{equation}\label{normform11}
  N_{\K/\Q}(L({\bf X})) = \prod_{j=1}^{k} (\alpha_1^{(j)}X_1+\dots+ \alpha_s^{(j)}X_s) = t,
\end{equation}
where $0\not=t\in \Z$, which solutions are searched in $\Z$. Notice that the polynomial $ N_{\K/\Q}(L({\bf X}))$ is invariant against conjugation, thus, it has rational integer coefficients.

Now we are in the position to state our Mahler-type result on the solutions of \eqref{normform11}.

\begin{theorem} \label{normformsol}
  Let $({\bf x}_n)=((x_{n1},\dots,x_{ns}))$ be a sequence of solutions of \eqref{normform11}, including infinitely many different ones. Let $1\le j \le s$ be fixed and $g\ge 2$. If $(x_{nj})$ is not ultimately zero then the infinite word $(|x_{1j}|)_g(|x_{2j}|)_g\ldots$ is not periodic.
\end{theorem}

{\bf Outline of the proof} By a deep theorem of W.M. Schmidt there exist a finite set $\mathcal{A}\subset\Z_{\K}$ such that
 $$
  \alpha_1x_{n1}+\dots+\alpha_sx_{ns} = \mu u_n
  $$
  with $\mu \in \mathcal{A}$ and with a unit $u_n\in \Z_{\K}$. Taking conjugates we obtain the system of linear equations
  $$
  \alpha_1^{(i)}x_{n1}+\dots+\alpha_s^{(i)}x_{ns} = \mu ^{(i)}u_n^{(i)}, i=1,\dots,k,
  $$
  which implies

  \newpage
  $$
    x_{nj} = \nu_1u_n^{(1)}+\dots+\nu_ku_n^{(k)}
  $$
with some constants $\nu_i$ belonging to the normal closure of $\K$. The assumption $(x_{nj})$ is not ultimately zero implies that $(x_{nj})$ is not bounded. Now we can apply Theorem 1.$\Box$
\vspace{0.5cm}

\begin{Cor} Let $g\ge 2$ be an integer. There are only finitely many $g$-repunits among the solutions of \eqref{normform11}.
\end{Cor}

\newpage

\begin{rem}
   If $\K$ is a real quadratic number field \eqref{normform11} is called Pell equation, which solutions can be expressed by the union of finitely many linear recursive sequences. In this case Theorem \ref{normformsol} is included implicitly in Theorem 1 of Barat, Frougny and Peth\H{o}.

   Gy\H{o}ry, Mignotte and Shorey, 1990 proved with the notation of Theorem \ref{normformsol} that if the set of the $j$-th coordinate of the solutions of \eqref{normform11} is not bounded then the greatest prime factor of them tends to infinity. Our Theorem \ref{normformsol} shows that their assumption always holds if \eqref{normform11} has infinitely many solutions, which $j$-th coordinates is non-zero.
\end{rem}

\newpage
{\bf 4. Families of GNS}

B. Kov\'acs, 1981: If $\K=\Q$ then for any $\gamma\in \Z_{\LL}$ there exists $N_1=N_1(\gamma)$ such that $(\gamma-m, \{0,1,\ldots,N_{\LL/\Q}(\gamma-m)\})$ {\bf is} a GNS in $\Z_{\LL}$ for all $m\ge N_1$. Moreover there exists $N_2=N_2(\gamma)$ such that $(\gamma+m, \{0,1,\ldots,N_{\LL/\Q}(\gamma+m)\})$ {\bf is not} a GNS in $\Z_{\LL}$ for all $m\ge N_2$.

Refinements by Akiyama and Rao, Scheicher and Thuswaldner. The proofs are based on the principle: Denote by $p(x)$ the minimal polynomial of $\gamma$. For $m\in \N$ we have $p(\mp m)=N_{\LL/\Q}(\gamma\pm m)$. If $m$ is large enough then $p(m)$ is dominating among the coefficients of $p(x+m)$ and $(\gamma-m,\{0,1,\ldots,p(m)-1\})$ is a GNS, while $|p(-m)|\in \{0,1,\ldots,|p(-m-1)|-1\}$, hence $(\gamma+m,\{0,1,\ldots,|p(-m)|-1\})$ is not a GNS.

In relative extensions we does not have natural ordering of the elements of the base field! A.P.and  Thuswaldner, 2018 found a way to overcome this difficulty. 

Let $\mathbb{K}$ be a number field of degree $k$. Let $\mathcal{F}$ be a bounded fundamental domain for the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$, {\it i.e.}, a set that satisfies $\mathbb{R}^k=\mathcal{F}+\mathbb{Z}^k$ without overlaps. Let $\OO$ be an order in $\Z_{\K}$, $\omega_1=1,\omega_2,\ldots,\omega_k$ be a $\mathbb{Z}$-basis of $\mathcal{O}$ and let $\alpha\in\mathcal{O}$ be given. Define
\begin{equation}\label{eq:DF}
D_{\mathcal{F},\alpha} = \Big\{\tau\in \mathcal{O} \; : \; \frac{\tau}{\alpha} = \sum_{j=1}^{k}r_j \omega_j, \,(r_1,\ldots, r_k) \in \mathcal{F} \Big\}.
\end{equation}

\begin{Le}
  
$D_{\mathcal{F},\alpha}$ is a complete residue system modulo $\alpha$.
\end{Le}

\newpage
Set $\mathbf{e}_1=(1,0,\ldots,0)\in \mathbb{R}^k$. ${\rm int}_+$ is the interior taken w.r.t.\ the subspace topology on $\{(r_1,\ldots, r_k) \in \mathbb{R}^k\;:\; r_1 \ge 0\}$.\\[0,5ex]

\begin{theorem}\label{th:newKovacs}
Let $\mathbb{K}$ be a number field of degree $k$ and let $\mathcal{O}$ be an order in $\mathbb{K}$. Let a bounded fundamental domain $\mathcal{F}$ for the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$ be given. Suppose that\\[0,5ex]
$\bullet $ $\mathbf{0} \in {\rm int}(\mathcal{F} \cup (\mathcal{F}-\mathbf{e}_1))$ and \\[0,5ex]
$\bullet$ $\mathbf{0} \in {\rm int}_+(\mathcal{F})$.\\
Let $\LL$ be a finite extension of $\K$ and $\gamma\in \Z_{\LL}$. Then there is $\eta > 0$ such that $(\gamma+ \alpha,D_{\mathcal{F},N_{\LL/\Q}(\gamma+\alpha)})$ is a GNS whenever $\alpha = m_1\omega_1 + \dots + m_k \omega_k\in \mathcal{O}$ satisfies $\max\{1,|m_2|,\ldots,|m_k|\}< \eta m_1 $.
\end{theorem}

\newpage
\begin{rem}
Note that this implies that for each bounded fundamental domain $\mathcal{F}$ satisfying 
\\[0,5ex]
$\bullet$ $\mathbf{0} \in {\rm int}(\mathcal{F} \cup (\mathcal{F}-\mathbf{e}_1))$ and\\[0,5ex]
$\bullet$ $\mathbf{0} \in {\rm int}_+(\mathcal{F})$.\\
the family $\mathcal{G}_\mathcal{F}$ contains infinitely many GNS.
\end{rem}

\newpage
\begin{Cor}\label{cor:newKovacs}
Let $\mathbb{K}$ be a number field of degree $k$ and let $\mathcal{O}$ be an order in $\mathbb{K}$. Let a bounded fundamental domain $\mathcal{F}$ for the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$ be given such that $\mathbf{0} \in {\rm int}(\mathcal{F})$. Let $\LL$ be a finite extension of $\K$ and $\gamma\in \Z_{\LL}$. Then there is $\eta > 0$ such that $(\gamma+ \alpha,D_{\mathcal{F},N_{\LL/\Q}(\gamma+\alpha)})$ has the finiteness property whenever $\alpha = m_1\omega_1 + \dots + m_k \omega_k\in \mathcal{O}$ satisfies $\max\{1,|m_2|,\ldots,|m_k|\}< \eta |m_1|$.
\end{Cor}

\newpage
{\bf 4.2. Families on non-GNS}

\vspace{0.5cm}
\begin{theorem} \label{non-ECNSmain}
Let $\mathbb{K}$ be a number field and let $\mathcal{O}$ be an order in $\mathbb{K}$. Let a bounded fundamental domain $\mathcal{F}$ for the action of $\mathbb{Z}^k$ on $\mathbb{R}^k$ be given. Suppose that
$\mathbf{0} \in {\rm int}_{-}(\mathcal{F}-\mathbf{e}_1)$.
Let $\LL$ be a finite extension of $\K$ and $\gamma\in \Z_{\LL}$. There exists $M \in \mathbb{N}$ such that $(\gamma+m,D_{\mathcal{F},N_{\LL/\Q}(\gamma+m)})$ is not a GNS for $m\ge M$.
\end{theorem}

\newpage
{\bf 4.3. GNS in number fields}

\vspace{0.5cm}
\begin{Pro}[{Kov\'acs and Peth\H{o}, 1991}]
  Let $\mathcal{O}$ be an order in the algebraic number field $\mathbb{K}$. There exist $\alpha_1,\dots,\alpha_t \in \mathcal{O}$, $n_1,\dots,n_t \in \mathbb{Z}$, and $N_1,\dots,N_t$ finite subsets of $\mathbb{Z}$, which are all effectively computable, such that $(\alpha,\{0,1,\ldots,N_{\K/\Q}(\alpha)\})$ is a GNS in $\mathcal{O}$ if and only if $\alpha=\alpha_i-h$ for some integers $i,h$ with $1\le i \le t$ and either $h\ge n_i$ or $h\in N_i$.
\end{Pro}

\newpage

Peth\H{o} and Thuswaldner, 2018 proved that the relation between power integral bases and GNS is usually stronger, the theorem of Kov\'acs and Peth\H{o} describes a kind of ``boundary case'' {\it viz.} a case where $0\in\partial\mathcal{F}$.\\[0.5ex]

\begin{theorem} \label{thm:newKovacsPetho}
Let $\mathcal{O}$ be an order in the algebraic number field $\mathbb{K}$. Let $\mathcal{F}$ be a bounded fundamental domain for the action of $\mathbb{Z}$ on $\mathbb{R}$. If $0 \in {\rm int}(\mathcal{F})$ then all but finitely many generators of power integral bases of $\mathcal{O}$ form a basis for a GNS. Moreover, the exceptions are effectively computable.
\end{theorem}

Evertse, Gy\H{o}ry, Peth\H{o} and Thuswaldner, 2019 generalized to \'etale orders. 

Partial answer to Problem 1. 

\begin{theorem} \label{th:givendigitset}
Let $\OO$ be an effectively given \'{e}tale order,
and $\DD$ a given finite subset of $\Z$ containing $0$. Then there exist only finitely many, effectively computable $\gamma\in \OO$ such that $(\gamma,\DD)$ ia a GNS.
\end{theorem}

\begin{proof}
Let $\gamma\in \OO$ and $\DD\subset \Z$ be such that $(\gamma,\DD)$ is a GNS. The set $\DD$ has to be a complete residue system of $\OO$ modulo $\gamma$, which is only possible if $|N(\gamma)| = |\DD|$.
If there is no such $\gamma$ then we are done. Otherwise, if $\OO\oz\Qq =\K_1\times\ldots \times\K_{\ell}$ and $\K_h$ are either the rational or an imaginary quadratic number field for all $h=1,\ldots,\ell$ then there are only finitely many $\alpha$ with $|N(\gamma)| = |\DD|$ and our assertion holds again.

\newpage
We now assume that there are infinitely many $\gamma\in \OO$ such that $|N(\gamma)| = |\DD|$. If $(\gamma,\DD)$ is a GNS then there exist for all $\alpha \in \OO$ an integer $L$ and $d_i\in \DD, i=0,\ldots,L$ such that
  $$
  \alpha = \sum_{i=0}^{L} d_i \gamma^{i},
  $$
  hence $\OO$ is monogenic.
  By a deep theorem of Evertse and Gy\ubH{o}ry there exist only finitely many $\Z$-equiva\-lence classes of $\beta\in \OO$ such that $\OO=\Z[\beta]$. Hence there is such a $\beta$ and $u\in \Z$ with $\alpha=\beta+u$. For fixed $\beta$ there are only finitely many effectively computable $u\in \Z$ with $|N(\beta+u)|=|\DD|$, thus the assertion is proved.
\end{proof}

  \newpage
  {\bf 5. Integers with bounded number of non-zero digits}
  
  Let $g_1,g_2\ge 2$ be integers.\\[0.5ex]
$\bullet$ Senge and Straus, 1973: the  number  of  integers,  the  sum  of  whose  digits  in  each of  the  bases $g_1$ and $g_2$ lies  below  a  fixed  bound,  is  finite  if  and  only  if $g_1$ and $g_2$ are multiplicatively independent.\\[0.5ex]
$\bullet$ Stewart, 1980: gave an effective version.\\[0.5ex]
$\bullet$ Schlickewei, 1990: ineffective generalization to more than two bases.\\[0.5ex]
$\bullet$ Peth\H{o} and Tichy, 1993: generalization to numeration systems based on linear recursive sequences and to GNS.

\begin{theorem}\label{pethotichy}
  Let $[\LL:\Q]=\ell\ge 2$ and $\gamma\in \Z_{\LL}, \DD\subset \Z$ such that $\gamma^{\ell}\notin \Z$ and $\DD$ is a complete residue system modulo $\gamma$. Assume that $(\gamma, \DD)$ is a GNS in $\Z_{\LL}$. Denote $r_{\gamma}(\alpha)$ the number of non-zero digits in the representation of $\alpha\in \Z_{\LL}$ in $(\gamma,\DD)$. For any $c>0$ there are only finitely many $n\in \Z$ such that $r_{\gamma}(n)\le c$.
\end{theorem}

 \end{document} 
 
Characterization of integers including special pattern in their $g$-ary representations lead to interesting and hard diophantine problems. I mention here only the repunits and integers having bounded number of non-zero digits in two different bases.

Thanks to intensive work, we have now an appropriate overview on radix representations in and over algebraic number fields. Let $\{\gamma,D\}$ be a GNS, i.e. $\gamma$ the bases and $D$ the digit set of a radix representation in an algebraic number field $\mathbb{K}$. For $\alpha\in \mathbb{K}$ denote by $(\alpha)_{\gamma}$ the word of the digits of the representation of $\alpha$. We discuss in our talk following topics:
\begin{enumerate}
  \item Representations of rational integers. We show under mild condition that there are only finitely many rational integer $m$ with bounded number of non-zero characters in $(m)_{\gamma}$.
  \item We have analogous statement for the representations of units. The results are effective. An interesting problem is to establish all units in cubic and/or quartic number fields with three,four, five non-zero digits. To do the same for rational integers seems more challenging.
  \item Let $w=(1)_{\alpha}(\alpha)_{\gamma}(\alpha^2)_{\gamma}\ldots$. By generalizing a result of K. Mahler we proved that $w$ is not periodic. (In fact we proved much more.) Write $w=s_1s_2\ldots, \; s_j\in D$ and $A=\sum_{j=1}^{\infty} s_j\gamma^j$. It is natural to expect $A\notin \mathbb{K}$, which holds for $\mathbb{K}=\Q$. What about if $\mathbb{K}\not=\Q$?
  \item Consider the set $F$ of the first coordinates of the integer solutions of a norm form equation $N_{\mathbb{K}/\Q}(\underbar x) = m$. (In the quadratic case the norm form equation is a Pell equation, thus the elements of $F$ form a linear recursive sequence.) We show that there are only finitely many repunits with respect to any base $g\ge 2$, in $F$. Give an algorithm for the computation of all repunits in the set $F$.
  \item With the solutions of relative index form equations one can establish the bases of - up to equivalence - all power integer bases in a relative extension of a number field. We proved that the bases of the radix representations are special translates of bases of power integral bases. All these results are effective, but nobody performed yet the necessary computations in a given case.
  \item For given $D\subset \Z$ there is only finitely many $\gamma \in \mathbb{K}$ such that $\{\gamma,D\}$ is a GNS. This is trivial for $\mathbb{K}=\Q$, but quite involved in general. For some low degree (2,3,4) number fields establish all binary and decimal GNS.
\end{enumerate}

\newpage

   Drmota and Ska{\l}ba (1991) Let $p\ge 3$ be a prime number and let
   $$
   f(X) = X^p + a_{p-1}X^{p-1}+\ldots+a_1 X+a_0 \in \mathbb{Q}[X]
   $$
   be an irreducible polynomial over $\mathbb{Q}$ such that $a_0\not=\pm 1$ and $a_j\not=0$ for at least one $j$ in the range $1\le j \le p-1$. Then, its roots are multiplicatively independent.

   Dobrowolski, Lemma 2, (1979): If $\alpha$ is a non-zero algebraic integer of degree $n$, then either
   \noindent
   (i) $\alpha_i^r \not=\alpha_j^s$ for rational integers $r>s\ge 1, 1\le i,j\le n $ and
   \noindent
   (ii) $|\prod_{1\le i,j \le n} (\alpha_i^p -\alpha_j| \ge p^n$ for prime numbers $p$
    \noindent
    hold or $\alpha$ is a root of unity.

   E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith, 34 (1979), 391--401

   M. Drmota and M. Ska{\l}ba, On multiplicative and linear independence of polynomial roots, in: Contributions to General Algebra 7 (eds. D. Dorninger et al.), Hoelder–Pichler–Tempsky, Wien, Teubner (Stuttgart, 1991), pp. 127–-135.




