Estimates for polynomials in logarithms of some rational numbers

The Hermite-Pade approximations of the second type for algebra generated by a generalized Nikishin system of Markov functions corresponding to an infinite branching graph are investigated. Arithmetical applications of this construction are given. Namely, lower estimates for polynomials with integer coefficients in logarithms of some rational numbers are obtained. These estimates partially refine some known results obtained earlier by the Siegel method. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 179–194, 2005.