We consider the sequence of fractional parts $\left\{\xi {\alpha }^n\right\}$, $n=1,2,3,\cdots$, where $\alpha >1$ is a Pisot number and $\xi \in \mathbb{Q}\left(\alpha \right)$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ such that $n\ge 1$, let ${\chi }_1,...,{\chi }_n$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $(p-3)/2$, set$${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_p}$ be an algebraic closure of ${𝔽}_p$. Our main result is that if the characters ${\chi }_i$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series $\overline{f(T,{\theta }_{i,j})}$ are linearly independent over a certain...

Let $f$ be a primitive cusp form of weight at least 2, and let ${\rho }_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of ${\rho }_f$ to a decomposition group at $p$ is “upper triangular”. If in addition $f$ has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members of a non-CM...

We prove the last of five outstanding conjectures made by R. M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer β all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value of one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for β with magnitudes up to 5 + 1/25....

For the cyclotomic ${\mathbb{Z}}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G\left(k_{\infty }\right)$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G\left(k_{\infty }\right)$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G\left(k_{\infty }\right)$ becomes a nonabelian metacyclic pro-$2$-group. We also calculate Iwasawa theoretically the Galois groups of $2$-class field towers of certain cyclotomic $2$-extensions.