On a variant of Pillai's problem with transcendental numbers (Q2151125)

For complex numbers \(\alpha\), \(\beta\) and \(0 < x \in \mathbb{R}\) put \[ T_{\alpha, \beta} (x) = \# \{ (n,m) \in \mathbb{N}^2 : \vert \alpha^n - \beta^m \vert \le x \}\ .\] A result of \textit{S. S. Pillai} [J. Indian Math. Soc. 19, 1--11 (1931; Zbl 0001.26802)] yields that for positive integers \(1 < \alpha, \beta\) one has the asymptotics \[ T_{\alpha, \beta} (x) \sim \frac {(\log x)^2} {\log \vert \alpha\vert \cdot \log \vert \beta\vert }\,, \quad \text{as } x \to \infty,\] and the authors ask whether this also holds for arbitrary, multiplicatively independent complex numbers with \(\vert \alpha\vert >1\) and \(\vert \beta\vert >1\). If \(\alpha\) and \(\beta\) are both algebraic numbers, an affirmative answer can be given with the methods from a recent paper of the authors [Res. Number Theory 7, No. 2, Paper No. 24, 12 p. (2021; Zbl 1473.11079)], but the question remained open if at least one of the numbers is transcendental. Using an extremely well approximable Liouville number the authors give an example for which the above asymptotics fails.\par For a real number \(\xi\), let \(\mu(\xi)\) denote the irrationality exponent of \(\xi\), i.e. the supremum of all \(\lambda\) for which the inequality \(0 < \bigl\vert \xi - \frac pq \bigr\vert < q^{-\lambda}\) has infinitely many solutions \((p,q) \in \mathbb{Z} \times \mathbb{N}\).\par The main result of this paper is that for \(2 \le \mu \bigl( \frac {\log \vert \alpha\vert } {\log \vert \beta\vert } \bigr) < \infty\) one has \[ T_{\alpha, \beta} (x) = \frac {(\log x)^2} {\log \vert \alpha\vert \cdot \log \vert \beta\vert } + O(\log x \cdot \log \log x) \quad \text{for } x \text{ large enough.}\] Consequently, the exceptions to the investigated asymptotics belong to a set of Lebesgue measure zero.