Exceptional parameters of linear mod one transformations and fractional parts \(\{ \xi (p/q)^n \}\) (Q2018264)

Given a linear mod one transformation \(f_{\beta,\alpha}(x)=\{\beta x+\alpha\}\) with \(\beta>1\) and \(0\leq\alpha<1\) real numbers, let \(S_{\beta, \alpha}\) be the set of all \(x\in[0,1)\) such that \(0\leq f^n_{\beta,\alpha}(x)<1\) for all \(n\geq 0\). For each fixed slope \(\beta>1\), define \(E_{\beta}\) to be the set of intercepts \(\alpha\in[0,1)\) such that \(S_{\beta, \alpha}\) is an infinite set. The motivation to study the set \(E_{\beta}\) mainly comes from Mahler's Z-number problem as showed in [\textit{L. Flatto} et al., Acta Arith. 70, No. 2, 125--147 (1995; Zbl 0821.11038)]. In particular, Flatto et al. conjectured that \(E_{\beta}\) is non-empty and has Lebesgue measure equal to zero. This was proved by \textit{Y. Bugeaud} in [Acta Arith. 114, No. 4, 301--311 (2004; Zbl 1061.11041)], who also showed that \(E_{\beta}\) is uncountable, not closed and proposed the question of determining its Hausdorff dimension. In this paper, the author completely settles Bugeaud's question proving that \(E_{\beta}\) has Hausdorff dimension zero. The proof of the main result is based on some properties of power series whose coefficients are Sturmian words. This leads the author to consider thinner measures than the \(s\)-dimensional Hausdorff measure and so to pose the question of finding an exact dimension function for \(E_{\beta}\).