On simultaneous approximation of algebraic numbers

Let Gamma subset of Q(-x) be a finitely generated multiplicative group of algebraic numbers. Let alpha(1), . . ., is an element of Q(-x) be algebraic numbers which are Q-linearly independent and let epsilon > 0 be a given real number. One of the main results that we prove in this article is as follows: There exist only finitely many tuples (u, q, p(1), . . ., p(r)) is an element of Gamma x Z(r+1) with d = [Q(u) : Q] for some integer d >= 1 satisfying vertical bar alpha(i)qu vertical bar> 1, vertical bar alpha(i)qu vertical bar is not a pseudo-Pisot number for some integer i is an element of{1, ..., r} and 0 < vertical bar alpha(i)qu - p(j)vertical bar -< 1/H-epsilon(u)vertical bar q vertical bar(d/r+epsilon) for all integers j = 1, 2,.,., r, where. H(u) is the absolute Weil height. In particular, when r = 1, this result was proved by Corvaja and Zannier in [Acta Math. 193 (2004), 175-191]. As an application of our result, we also prove a transcendence criterion which generalizes a result of Hancl, Kolouch, Pulcerova, and Stepnicka in [Czech. Math. J. 62 (2012), no. 3, 613-623]. The proofs rely on the clever use of the subspace theorem and the underlying ideas from the work of Corvaja and Zannier.