On the rational approximation to linear combinations of powers

Abstract For a complex number x , ∥ x ∥ := min ⁡ { | x - m | : m ∈ ℤ } {\|x\|:=\min\{|x-m|:m\in\mathbb{Z}\}} . Let k ≥ 1 {k\geq 1} be an integer, and let K be a number field. Let α 1 , … , α k {\alpha_{1},\dots,\alpha_{k}} be algebraic numbers with | α i | ≥ 1 {|\alpha_{i}|\geq 1} and let d i {d_{i}} denotes the degree of α i {\alpha_{i}} for 1 ≤ i ≤ k {1\leq i\leq k} . Set d = d 1 + ⋯ + d k {d=d_{1}+\cdots+d_{k}} . In this article, we show that if the inequality 0 < ∥ λ 1 ⁢ q ⁢ α 1 n