We prove that the generalized Thue-Morse word ${𝐭}_{b,m}$ defined for $b\ge 2$ and $m\ge 1$ as ${𝐭}_{b,m}={\left(s_b\left(n\right)modm\right)}_{n=0}^{+\infty }$, where $s_b\left(n\right)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering antimorphisms $\Theta \in D_m$, we show that ${𝐭}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the generalisation of palindromic richness recently introduced by the author...

A simple Parry number is a real number $\beta \>1$ such that the Rényi expansion of $1$ is finite, of the form $d_{\beta }\left(1\right)=t_1\cdots t_m$. We study the palindromic structure of infinite aperiodic words $u_{\beta }$ that are the fixed point of a substitution associated with a simple Parry number $\beta$. It is shown that the word $u_{\beta }$ contains infinitely many palindromes if and only if $t_1=t_2=\cdots =t_{m-1}\ge t_m$. Numbers $\beta$ satisfying this condition are the so-called Pisot numbers. If $t_m=1$ then $u_{\beta }$ is an Arnoux-Rauzy word. We show that if $\beta$ is a confluent Pisot number then...

An infinite word is $S$-automatic if, for all $n\ge 0$, its $(n+1)$st letter is the output of a deterministic automaton fed with the representation of $n$ in the considered numeration system $S$. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for $d\ge 2$, we state that a multidimensional infinite word $x:{\mathbb{N}}^d\to \Sigma$ over a finite alphabet $\Sigma$ is $S$-automatic for some abstract...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...

Suppose $A$ is a set of non-negative integers with upper Banach density $\alpha$ (see definition below) and the upper Banach density of $A+A$ is less than $2\alpha$. We characterize the structure of $A+A$ by showing the following: There is a positive integer $g$ and a set $W$, which is the union of $\lceil 2\alpha g-1\rceil$ arithmetic sequences [We call a set of the form $a+d\mathbb{N}$ an arithmetic sequence of difference $d$ and call a set of the form $\{a,a+d,a+2d,...,a+kd\}$ an arithmetic progression of difference $d$. So an arithmetic progression is finite and an arithmetic...