On a variant of Pillai's problem with transcendental numbers

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