Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$. The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

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...

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...

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 $p\mathbb{Z}$.

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 $C$ be a smooth real quartic curve in ${\mathbb{P}}^2$. Suppose that $C$ has at least $3$ real branches $B_1,B_2,B_3$. Let $B=B_1\times B_2\times B_3$ and let $O\in B$. Let ${\tau }_O$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(ℝ\right)}^0$ of the set of real points of the jacobian of $C$, defined by letting ${\tau }_O\left(P\right)$ be the divisor class of the divisor $\sum P_i-O_i$. Then, ${\tau }_O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(ℝ\right)}^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real...