Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier
Avinash Kulkarni, Niki Myrto Mavraki, Khoa D. Nguyen · Transactions of the American Mathematical Society · 2017
For every complex number x x , let ‖ x ‖ Z := min { | x − m | : m ∈ Z } \Vert x\Vert _{\mathbb {Z}}:=\min \{|x-m|:\ m\in \mathbb {Z}\} . Let K K be a number field, let k ∈ N k\in \mathbb {N} , and let α 1 , … , α k \alpha _1,\ldots ,\alpha _k be non-zero algebraic numbers. In this paper, we completely solve the problem of the existence of θ ∈ ( 0 , 1 ) \theta \in (0,1) such that there are infinitely many tuples ( n , q 1 , … , q k ) (n,q_1,\ldots ,q_k) satisfying ‖ q 1 α 1 n