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Abstract: We give an effective lower bound for the number of non-zero digits among the first N digits of the expansion of an irrational algebraic number in an integer base. [Copyright &y& Elsevier]

Abstract: Let θ be a Pisot number less than 2, m a positive rational integer, and the set of the polynomials with coefficients in evaluated at θ. We give a lower bound for the greatest limit point of common differences of consecutive elements of . To cite this article: T. Zaïmi, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: In this Note, we construct for some normal surface X an object of whose homology sheaves are not strictly -invariant. This disproves the -connectivity conjecture of F. Morel. To cite this article: J. Ayoub, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Let (X,ω) be a compact complex Hermitian manifold, and let T⩾γ be a d-closed (1,1) almost positive current on X. A variant of Demailly''s regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents Tm with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C∞, the regularizing currents Tm can be chosen such that Tm⩾γ−C/m for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

Abstract: We give an elementary counter-example for a conjecture of Pineiro, Szpiro and Tucker about the uniform distribution of algebraic numbers with small height. To cite this article: P. Autissier, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]