Approximation of an algebraic number by products of rational numbers and units

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms

A family of Thue equations involving powers of units of the simplest cubic fields

E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms $F_n(X, Y )= X^3 -(n-1)X^2Y -(n+2)XY^2 -Y^3$ and the family of equations $F_n(X, Y )=\pm 1$, $n\in {\mathbf N}$. This family is associated to the family of the simplest cubic fields ${\mathbf Q}(\lambda)$ of D. Shanks, $\lambda$ being a root of $F_n(X,1)$. We introduce in this family a second parameter by replacing the roots of the minimal polynomial $F_n(X, 1)$ of $\lambda$ by the $a$-th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters $n$ and $a$.Comment: Expanded version (31p) of a paper to appear in the Journal de Th\'eorie des Nombres de Bordeau

Investigation of the Praesepe cluster. III. Radial velocity and binarity of the F5-K0 Klein-Wassink stars

Coravel observations of 103 F5-K0 stars in the Praesepe cluster yielded 24 spectroscopic binaries (3 are non-members), and 20 orbits were determined, with periods from 4 to 7400 days. Based on a complete sample in the colour range 0.40 < B-V < 0.80 (80 stars, including KW 244 = TX Cnc), the companion star fraction CSF = 0.45. The percentage of spectroscopic binaries with P < 1000d is 20% (16/80). The combined photometric and spectroscopic analysis showed that 12 among 18 single-lined spectroscopic binaries are located within the "single" star sequence in the (V,B-V) diagram and cannot be detected by the photometric analysis in the UBV system. In addition, seven photometrically analysed binaries were not detected with the radial velocity observations, but are confirmed members. The number of single:binary:triple stars is 47:30:3.Comment: 10 pages, 3 tables, 7 eps figures. Accepted for A&A. LaTe