On a variant of Pillai's problem with transcendental numbers

In this paper, we study the asymptotic formula for the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | \alpha^n - \beta^m | \leq x $ when $x$ goes to infinity. Here $\alpha, \beta$ are given multiplicatively independent complex numbers with $|\alpha| > 1$ and $|\beta|>1$.