On the rational approximation to linear combinations of powers

For a complex number $x$, $\Vert x\Vert:=\min\{|x-m|:m\in\mathbb{Z}\}$. Let $k\geq 1$ be an integer, and $K$ be a number field. Let $\alpha_1,\ldots,\alpha_k$ be algebraic numbers with $|\alpha_i|\geq 1$ and let $d_i$ denotes the degree of $\alpha_i$ for $1\leq i\leq k$. Set $d=d_1+\cdots+d_k$. In this article, we show that if the inequality $0<\Vert\lambda_1 q\alpha^n_1+\cdots+\lambda_k q\alpha^n_k\Vert<\frac{\theta^n}{q^{d+\varepsilon}}$ has infinitely many solutions in $(n, q,\lambda_1,\ldots,\lambda_k)\in \mathbb{N}^2\times (K^\times)^k$ with absolute logarithmic Weil height of $\lambda_i$ is small compared to $n$ and some $\theta\in (0,1)$, then, in particular, the tuple $(\lambda_1 q\alpha^n_1,\ldots, \lambda_k q\alpha^n_k)$ is pseudo-Pisot, and at least one of $\alpha_i$ is an algebraic integer. This result can be viewed as Roth's type theorem for linear combinations of powers of algebraic numbers over $\overline{\mathbb{Q}}$. The case $q=1$ was recently proved by Kulkarni, Mavraki, and Nguyen \cite{kul}, which is a generalization of Mahler's question proved in \cite{corv}. As a consequence of our result, we obtain the following generalization of this question: let $\alpha>1$ be an algebraic number with $d=[\mathbb{Q}(\alpha):\mathbb{Q}]$. For a given $\varepsilon>0$, if the inequality $$0<\Vert\lambda q\alpha^n\Vert<\frac{\theta^n}{q^{d+\varepsilon}}$$ has infinitely many solutions in the tuples $(n,q,\lambda)\in \mathbb{N}^2\times K^\times$ with absolute logarithmic Weil height of $\lambda$ is small compared to $n$ and $\theta\in (0,1)$, then some power of $\alpha$ is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.