On inhomogeneous extension of Thue-Roth's type inequality with moving targets

Abstract

Let $\Gamma \subset {\overline{\mathbb{Q}}}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\beta \in {\overline{\mathbb{Q}}}^{\times }$ be algebraic numbers with $\beta$ irrational. In this paper, we prove that there exist only finitely many triples $(u,q,p)\in \Gamma \times {\mathbb{Z}}^2$ with $d=[\mathbb{Q}(u):\mathbb{Q}]$ such that $$0<|\delta qu+\beta -p|<\frac{1}{H^{\epsilon }(u)q^{d+\epsilon }},$$ where $H(u)$ denotes the absolute Weil height. As an application of this result, we also prove a transcendence result, which states as follows: Let $\alpha >1$ be a real number. Let $\beta$ be an algebraic irrational and $\lambda$ be a non-zero real algebraic number. For a given real number $\epsilon >0$, if there are infinitely many natural numbers $n$ for which $||\lambda {\alpha }^n+\beta ||<2^{-\epsilon n}$ holds true, then $\alpha$ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha$ and $\beta$ both are algebraic satisfying same conditions, then a particular result of Kulkarni, Mavraki and Nguyen, proved in [3] asserts that ${\alpha }^d$ is a Pisot number. When $\beta$ is algebraic irrational, our result implies that no algebraic number $\alpha$ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [6]. The proof of our results uses the Subspace Theorem based on the idea of Corvaja and Zannier [2] together with various modification play a crucial role in the proof.

arXiv admin note: text overlap with arXiv:2001.00386

References in corpus (1)

• Irrationality of growth constants associated with polynomial recursions