On simultaneous approximation of algebraic numbers

Abstract

Let Γ ⊂ ℚ × be a finitely generated multiplicative group of algebraic numbers. Let 𝛼 1 , … , 𝛼 𝑟 ∈ ℚ × be algebraic numbers which are ℚ-linearly independent and let 𝜖 > 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 tuplesPisot number for some integer 𝑖 ∈ {1, … , 𝑟} and 0 < |𝛼 𝑗 𝑞𝑢 − 𝑝 𝑗 | < 1 𝐻 𝜖 (𝑢)|𝑞| 𝑑 𝑟 +𝜀for 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 Hančl, Kolouch, Pulcerová, and Štěpnička 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.M S C 2 0 2 0 11J68 (primary), 11J87 (secondary)