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A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm

Jun-ichi Tamura (1992)

Acta Arithmetica

In this paper, we give transcendental numbers φ and ψ such that (i) both φ and ψ have explicit g-adic expansions, and simultaneously, (ii) the vector t ( φ , ψ ) has an explicit expression in the Jacobi-Perron algorithm (cf. Theorem 1). Our results can be regarded as a higher-dimensional version of some of the results in [1]-[5] (see also [6]-[8], [10], [11]). The numbers φ and ψ have some connection with algebraic numbers with minimal polynomials x³ - kx² - lx - 1 satisfying (1.1) k ≥ l ≥0, k + l ≥ 2 (k,l...

Conjecture de Littlewood et récurrences linéaires

Bernard de Mathan (2003)

Journal de théorie des nombres de Bordeaux

Ce travail est essentiellement consacré à la construction d’exemples effectifs de couples ( α , β ) de nombres réels à constantes de Markov finies, tels que 1 , α et β soient 𝐙 -linéairement indépendants, et satisfaisant à la conjecture de Littlewood.

Diophantine approximation with partial sums of power series

Bruce C. Berndt, Sun Kim, M. Tip Phaovibul, Alexandru Zaharescu (2013)

Acta Arithmetica

We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.

Linear independence of linear forms in polylogarithms

Raffaele Marcovecchio (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

For x ∈ ℂ , | x | < 1 , s ∈ ℕ , let Li s ( x ) be the s -th polylogarithm of x . We prove that for any non-zero algebraic number α such that | α | < 1 , the ℚ ( α ) -vector space spanned by 1 , Li 1 ( α ) , Li 2 ( α ) , ⋯ has infinite dimension. This result extends a previous one by Rivoal for rational α . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

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