Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja–Zannier

For every complex number xx, let ‖x‖Z:=min{|x−m|: m∈Z}\Vert x\Vert _{\mathbb {Z}}:=\min \{|x-m|:\ m\in \mathbb {Z}\}. Let KK be a number field, let k∈Nk\in \mathbb {N}, and let α1,…,αk\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,q1,…,qk)(n,q_1,\ldots ,q_k) satisfying ‖q1α1n+⋯+qkαkn‖Z>θn\Vert q_1\alpha _1^n+\cdots +q_k\alpha _k^n\Vert _{\mathbb {Z}}>\theta ^n where n∈Nn\in \mathbb {N} and q1,…,qk∈K∗q_1,\ldots ,q_k\in K^* have small logarithmic height compared to nn. In the special case when q1,…,qkq_1,\ldots ,q_k have the form qi=qciq_i=qc_i for fixed c1,…,ckc_1,\ldots ,c_k, our work yields results on algebraic approximations of c1α1n+⋯+ckαknc_1\alpha _1^n+\cdots +c_k\alpha _k^n of the form mq\frac {m}{q} with m∈Zm\in \mathbb {Z} and q∈K∗q\in K^* (where qq has small logarithmic height compared to nn). Various results on linear recurrence sequences also follow as an immediate consequence. The case where k=1k=1 and q1q_1 is essentially a rational integer was obtained by Corvaja and Zannier and settled a long-standing question of Mahler. The use of the Subspace Theorem based on work of Corvaja–Zannier, together with several modifications, plays an important role in the proof of our results.