A note on Diophantine fractals for \(\alpha \)-Lüroth systems (Q2855527)

This paper considers the \(\alpha\)-Lüroth map \(L_\alpha:[0,1] \rightarrow [0,1]\) where \(\alpha =\{A_n:n \in \mathbb{N}\}\) is a certain type of countable partition of \([0,1]\) into subintervals. Each \(x \in [0,1]\) determines a sequence \(\{l_k\}_{k \geq 1}\) where \(L_{\alpha}^{k-1}(x) \in A_{l_k}\) and each \(x\) has a unique \(\alpha\)-Lüroth expansion determined by \(\{l_k\}_{k \geq 1}\). Under an assumption that the tails of \(\alpha\) satisfy a certain power law, the author proves some results about the Hausdorff dimensions of some subsets of \([0,1]\). For example, for NEWLINE\[NEWLINEG_N^{(\alpha)} = \{x \in [0,1]: l_i(x)>N \text{ for all } i \in \mathbb{N}\},NEWLINE\]NEWLINE NEWLINE\[NEWLINEG_\infty^{(\alpha)} = \{x \in [0,1]: \lim_{n \rightarrow \infty}l_n(x)= \infty\},NEWLINE\]NEWLINE the author calculates \(\lim_{N \rightarrow \infty}\dim_H(G_N^{(\alpha)} )\) and \(\dim_H(G_\infty^{(\alpha)} )\). More on the \(\alpha\)-Lüroth expansion can be found in [\textit{J. Barrionuevo} et al., Acta Arith. 74, No. 4, 311--327 (1996; Zbl 0848.11039)].