Let $\Theta =({\theta }_1,{\theta }_2,{\theta }_3)\in {ℝ}^3$. Suppose that $1,{\theta }_1,{\theta }_2,{\theta }_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$\begin{array}{cc}\hfill \alpha \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; sup}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \},\hfill \\ \hfill \beta \left(\Theta \right)& =sup\{\gamma >0:\underset{t\to +\infty }{lim\; inf}{t}^{\gamma }{\psi }_{\Theta }\left(t\right)<+\infty \}\hfill \end{array}$$ we prove $$\beta \left(\Theta \right)\ge \frac{1}{2}\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}+\sqrt{{\left(\frac{\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}\right)}^2+\frac{4\alpha \left(\Theta \right)}{1-\alpha \left(\Theta \right)}}\right)\alpha \left(\Theta \right).$$

Let $\xi$ be a real number and let $n$ be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents $w_n\left(\xi \right)$ and $w_n^{*}\left(\xi \right)$ defined by Mahler and Koksma. We calculate their six values when $n=2$ and $\xi$ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction $\xi$ by quadratic surds.

On montre que les exposants de Lyapunov de l’algorithme de Jacobi-Perron, en dimension $d$ quelconque, sont tous différents et que la somme des exposants extrêmes est strictement positive. En particulier, si $d=2$, le deuxième exposant est strictement négatif.

Dans cet article nous présentons la théorie des équations différentielles $p$-adiques et ses applications concernant le théorème de finitude de la cohomologie $p$-adique d’une variété affine et le théorème de la monodromie $p$-adique des représentations galoisiennes locales.

We prove the rationality of the Łojasiewicz exponent for p-adic semi-algebraic functions without compactness hypothesis. In the parametric case, we show that the parameter space can be divided into a finite number of semi-algebraic sets on each of which the Łojasiewicz exponent is constant.

We use hyperbolic geometry to study the limiting behavior of the average number of ways of expressing a number as the sum of two coprime squares. An alternative viewpoint using analytic number theory is also given.

In this article we formalize some number theoretical algorithms, Euclidean Algorithm and Extended Euclidean Algorithm [9]. Besides the a gcd b, Extended Euclidean Algorithm can calculate a pair of two integers (x, y) that holds ax + by = a gcd b. In addition, we formalize an algorithm that can compute a solution of the Chinese remainder theorem by using Extended Euclidean Algorithm. Our aim is to support the implementation of number theoretic tools. Our formalization of those algorithms is based...