Nombres de Liouville et nombres normauxLiouville numbers and normal numbers

We prove that there exist Liouville numbers which are normal, as well as Liouville numbers which are non-normal to any base. To cite this article: Y. Bugeaud, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 117–120.     

Approximation by Algebraic Integers and Hausdorff Dimension

The paper computes the Hausdorff dimension of sets of real numberswhich are close to infinitely many real algebraic integers ofbounded degree. It also investigates the distribution of realalgebraic integers of bounded degree, which are proved to beevenly spaced.     

Algebraic Numbers Close to 1 in Non-Archimedean Metrics

Let 1 be an algebraic number with relatively small height. Recently, many authors, including Amoroso, Dubickas, Mignotte and Waldschmidt, stated sharp lower bounds for the quantity | – 1|. Here, we provide a p-adic analogue of their results. For instance, we give an upper bound for the absolute value of the norm of – 1, and we show that our estimate is rather sharp in terms of the degree of . Further, we discuss a generalization in several variables of our result.     

On the diophantine equation (x − 1)(y − 1) = (z − 1)

Let k ≥ 3 be an integer. We study the possible existence of pairs of distinct positive integers (a, b) such that any of the three numbers a + 1, b + 1, and ab + 1 is a k-th power. We further investigate several related questions.     

On the diophantine equation

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation . Moreover, we prove that the diophantine equation , , , , , gcd, , has only finitely many solutions, all of which satisfying .

     

On Some Exponential Diophantine Equations

 We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000)     

On transfer inequalities in Diophantine approximation, II

Let Θ be a point in R ⁿ . We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(R ⁿ⁺¹). It turns out that our convex bodies in degree d are the dth compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables us both to split and to refine the classical Khintchine transference principle.     

On U m-numbers with small transcendence measure

We provide a new, effective method for constructing real U _m ^- numbers ξ with sharp upper bounds for w_n ^*(ξ), where n = 1,...,m_1. This revised version was published online in August 2006 with corrections to the Cover Date.