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 talk, I will 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
$$
0<|\delta qu+\beta-p|<\frac{1}{H^\vareps
$$
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 $\varepsilon >0$, if there are infinitely many
natural numbers $n$ for which $||\lambda\alpha^n+\beta|| < 2^{-
\varepsilon n}$ holds true, then $\alpha$ is transcendental, where
$||x||$ denotes the distance from its nearest integer.
Google Meet Link: meet.google.com/rpj-qpwn-ows