We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^A$ and height at most $A{p}^A$, where $A$ is an effective constant that only depends on...

Soient $G$ une variété de groupe définie sur le corps $\bar{\mathbf{Q}}$ des nombres algébriques, et $\phi :{\mathbf{C}}^n\to G_{\mathbf{C}}$ un sous-groupe à $n$ paramètres de $G$, de dimension algébrique $d$. Nous nous proposons de majorer le rang $\ell$ (sur $\mathbf{Z}$) des sous-groupes $\Gamma$ de ${\mathbf{C}}^n$ dont l’image par $\phi$ est contenue dans le groupe $G_{\bar{\mathbf{Q}}}$ des points algébriques de $G$.E. Bombieri et S. Lang ont déjà obtenu de telles majorations, en supposant que les points de $\Gamma$ sont très bien distribués : pour $d\ge n+1$, on a $\ell \le n^2+3n$ pour des variétés linéaires, et $\ell \le 2n^2+4n$ pour des variétés abéliennes .Nous...

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ and their associated approximation constants. As an application, explicit construction of real numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

We show that Y. Cheung’s general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...

This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos' results on~irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

We prove a new lower bound for the height of points on a subvariety $V$ of a multiplicative torus, which lie outside the union of torsion subvarieties of $V$. Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an $\epsilon$ the sharpest conjectures that can be formulated....