A point $({\xi }_1,{\xi }_2)$ with coordinates in a subfield of $ℝ$ of transcendence degree one over $\mathbb{Q}$, with $1,{\xi }_1,{\xi }_2$ linearly independent over $\mathbb{Q}$, may have a uniform exponent of approximation by elements of ${\mathbb{Q}}^2$ that is strictly larger than the lower bound $1/2$ given by Dirichlet’s box principle. This appeared as a surprise, in connection to work of Davenport and Schmidt, for points of the parabola $\{(\xi ,{\xi }^2);\xi \in ℝ\}$. The goal of this paper is to show that this phenomenon extends to all real conics defined over $\mathbb{Q}$, and that the largest exponent of...

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions.Improved bounds for the Chebyshev functions in arithmetic progressions $\theta (k,l;x)$ and $\psi (k,l;x)$ for $k=1,3,4,6$ are also presented....

We give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give a few examples of algebraic Laurent series for which...

Let $X\subset {ℝ}^n$ be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of $X$ that do not lie on some connected semialgebraic curve contained in $X$.

It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $$r_i=\frac{a_i}{b_i}$$ for some integers a i, b i.⊎ for all $$i,0⩽\left|a_i\right|⩽b_i⩽{\left(\frac{{32}^{1/2}⌈lo{g}_{2}n⌉}{\epsilon }\right)}^{2⌈lo{g}_{2}n⌉}$$ . One consequence of this...

We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.

On sait (Cobham) qu’une suite $2$- et $3$-automatique est une suite rationnelle. Une question de Loxton et van der Poorten étend ce résultat au cas $2$- et $3$-régulier. On montre dans cet article que, si une suite vérifie une récurrence $2$- et $3$-mahlérienne d’ordre un, elle est rationnelle.