On a variant of Pillai's problem with transcendental numbers

Abstract

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

This is not the final published version of the paper. Acta Math. Hungar. (2022)