A transference principle for simultaneous rational approximation

We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.Comment: 15 page

SS-unit equation in two variables and Pad\'{e} approximations

In this article, we use Pad\'{e} approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Pad\'{e} approximants with a simple argument relying on Mahler measure and on the local height, we refine the bound due to J.-H. Evertse.Comment: 13 page

Pad\'e approximation for a class of hypergeometric functions and parametric geometry of numbers

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Pad\'e approximations by using a formal method and show that the associated sequences satisfy a Poincar\'e-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an \emph{effective} version of the Poincar\'e-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers.Comment: 33 pages, 1 table, minor corrections, references update

Parametric geometry of numbers over a number field and extension of scalars

International audienceThe parametric geometry of numbers of Schmidt and Summerer deals with rational approximation to points in $\mathbb{R}^n$. We extend this theory to a number field $K$ and its completion $K_w$ at a place $w$ in order to treat approximation over $K$ to points in $K_w^n$. As a consequence, we find that exponents of approximation over $\mathbb{Q}$ in $\mathbb{R}^n$ have the same spectrum as their generalizations over $K$ in $K_w^n$. When $w$ has relative degree $1$ over a place $\ell$ of $\mathbb{Q}$, we further relate approximation over $K$ to a point $\xi$ in $K_w^n$, to approximation over $\mathbb{Q}$ to a point $\Xi$ in $\mathbb{Q}_{\ell}^{nd}$ obtained from $\xi$ by extension of scalars, where $d$ is the degree of $K$ over $\mathbb{Q}$. By combination with a result of P. Bel, this allows us to construct algebraic curves in $\mathbb{R}^{3d}$ defined over $\mathbb{Q}$, of degree $2d$, containing points that are very singular with respect to rational approximation