Generalizing a result of Pourchet, we show that, if $\alpha ,\beta$ are power sums over $\mathbb{Q}$ satisfying suitable necessary assumptions, the length of the continued fraction for $\alpha \left(n\right)/\beta \left(n\right)$ tends to infinity as $n\to \infty$. This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers $\alpha \left(n\right)/\beta \left(n\right)$, $n\in \mathbb{N}$.

We prove the existence of a limit distribution of the normalized well-distribution measure $W\left(E_N\right)/\sqrt{N}$ (as $N\to \infty$) for random binary sequences $E_N$, by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.

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...