On a variant of Pillai's problem with transcendental numbers

TL;DR

This paper investigates the asymptotic number of solutions to an inequality involving powers of multiplicatively independent complex numbers, extending Pillai's problem to transcendental number contexts.

Contribution

It provides new asymptotic estimates for solutions to a variant of Pillai's problem with transcendental numbers, a topic not extensively explored before.

Findings

Derived asymptotic formulas for the number of solutions as x tends to infinity

Extended Pillai's problem to complex, transcendental settings

Established bounds for solutions involving multiplicatively independent 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 | \leq x $ when $x$ tends to infinity. Here $\alpha, \beta$ are given multiplicatively independent complex numbers with $|\alpha| > 1$ and $|\beta|>1$.