Let \(\Gamma\in\overline{\mathbb Q}^\times\) be a finitely generated multiplicative group of algebraic numbers. Let \(\delta\) be a non-zero algebraic number, \(\beta\in (0,1)\) be an algebraic irrational and \(\varepsilon >0\) be a fixed real number. Then the author proves that there exists only finitely many triples \((u,q,p)\in\Gamma\times\mathbb Z^2\) with \(d=[\mathbb Q(u):\mathbb Q]\) such that \(\mid\delta qu\mid >1\) and \[ 0<\mid\delta qu+\beta-p\mid <\frac 1{H^\varepsilon(u)q^{d+\varepsilon}}. \] As a consequence the author proves that if \(\alpha\) is a real number, \(\beta\) is an algebraic irrational and \(\lambda\) is non-zero real algebraic number then for a given real number \(\varepsilon >0\) if there are infinitely many natural numbers \(n\) for which \(\parallel \lambda\alpha^n+\beta\parallel <2^{-\varepsilon n}\) holds true, then \(\alpha\) is transcendental.