Given a large prime number $p$ and a rational function $r\left(X\right)$ defined over ${𝔽}_p=\mathbb{Z}/p\mathbb{Z}$, we investigate the size of the set $\left\{x\in {𝔽}_p:\tilde{r}\left(x\right)\>\tilde{r}(x+1)\right.$, where $\tilde{r}\left(x\right)$ and $\tilde{r}(x+1)$ denote the least positive representatives of $r\left(x\right)$ and $r(x+1)$ in $\mathbb{Z}$ modulo $p\mathbb{Z}$.

Let $M\left(r\right)$ denote the set of real numbers $\alpha$ whose base-$r$ digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the set $M\left(r\right)$ is closed under multiplication by rational numbers, but not closed under addition.

We investigate in a geometrical way the point sets of  $ℝ$  obtained by the  $\beta$-numeration that are the  $\beta$-integers  ${\mathbb{Z}}_{\beta }\subset \mathbb{Z}\left[\beta \right]$  where  $\beta$  is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the  $\beta$-numeration, allowing to lift up the  $\beta$-integers to some points of the lattice  ${\mathbb{Z}}^m$  ($m=$  degree of  $\beta$) lying about the dominant eigenspace of the companion matrix of  $\beta$ . When  $\beta$  is in particular a Pisot number, this framework gives another proof of the fact...

Let $f\left(x\right)$ be a power series ${\sum }_{n\ge 1}\zeta \left(n\right)x^{e\left(n\right)}$, where $\left(e\right(n\left)\right)$ is a strictly increasing linear recurrence sequence of non-negative integers, and $\left(\zeta \right(n\left)\right)$ a sequence of roots of unity in ${\overline{\mathbb{Q}}}_p$ satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ${\mathbb{Q}}_p$ of the elements $f\left({\alpha }_1\right),...,$ $f\left({\alpha }_t\right)$ from ${ℂ}_p$ in terms of the distinct ${\alpha }_1,...,{\alpha }_t\in {\mathbb{Q}}_p$ satisfying $0\<|{\alpha }_{\tau }{|}_p\<1$ for $\tau =1,...,t$. A striking application of our basic result says that, in the case $e\left(n\right)=n$, the set $\left\{f\left(\alpha \right)\right|\alpha \in {\mathbb{Q}}_p,0\<|\alpha {|}_p\<1\}$ is algebraically independent over ${\mathbb{Q}}_p$ if...

The present paper shows that for any $s$ sequences of real numbers, each with infinitely many distinct elements, ${\lambda }_n^j$, j=1,...,s, the rational combinations of $x_1^{{\lambda }_{m_1}^1}{x}_2^{{\lambda }_{m_2}^2}...x_s^{{\lambda }_{m_s}^s}$ are always dense in $C_{I^s}$.

For an arbitrary (not totally real) number field $K$ of degree $\ge 3$, we ask how many perfect powers ${\gamma }^p$ of algebraic integers $\gamma$ in $K$ exist, such that $\mu \left(\tau \left({\gamma }^p\right)\right)\le X$ for each embedding $\tau$ of $K$ into the complex field. ($X$ a large real parameter, $p\ge 2$ a fixed integer, and $\mu \left(z\right)=max\left(\right|\mathrm{Re}\left(z\right)|,|\mathrm{Im}\left(z\right)\left|\right)$ for any complex $z$.) This quantity is evaluated asymptotically in the form $c_{p,K}{X}^{n/p}+R_{p,K}\left(X\right)$, with sharp estimates for the remainder $R_{p,K}\left(X\right)$. The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result...