Gaps in intervals of \(N\)-expansions (Q6546723)

The authors continue the work on so-called \(N\)-fractions (a type of continued fraction) done \Ntogether with \textit{J. de Jonge} et al. [Monatsh. Math. 198, No. 1, 79--119 (2022; Zbl 1494.11059)].\N\NThe abstract gives the main concepts in a nutshell:\N\N``For \(N\in \mathbb{N}_{\geq 2}\) and \(\alpha\in\mathbb{R}\) such that \(0<\alpha \leq\sqrt{N}-1\), the continued fraction map \(T_{\alpha}\,:\,[\alpha,\alpha+1]\rightarrow[\alpha,\alpha+1\rfloor\) is defned as \(T_{\alpha}(x):=N/x-d(x)\) where \N\(d\,:\,[\alpha,\alpha+1]\rightarrow\mathbb{N}\) is defined by \(d(x):=\lfloor N/x-\alpha\rfloor\).\N\NA maximal open interval \((a,b)\subset I_{\alpha}\) is called a \textit{gap} if \(I_{\alpha}\) if for almost every \(x\in I_{\alpha}\) there is an \(n_0(x)\in\mathbb{N}\) such that \(x_n\not\in (a,b)\) for all \(n\geq n_0\). In this paper all conditions are given in which \(I_{\alpha}\) is gapless. For \(\alpha=\sqrt{N}-1\) it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of \(N\).''\N\NThe layout of the paper is as follows:\N\N\S1. Introduction (\(5\) pages). Definitions and notations, background material (a.o. the paper where the definition of this type of expansions originally was given: [\textit{C. Kraaikamp} and \textit{N. Langeveld}, J. Math. Anal. Appl. 454, No. 1, 106--126 (2017; Zbl 1366.65002)].\N\NThe Theorems 1--4 are taken form this paper and the new result is:\N\NTheorem 5. All endpoints of gaps of \(I_{\alpha}\) are in the orbits of \(\alpha\) and \(\beta\).\N\N\S2. Gaps in \(I_{\alpha}\) when \(\alpha=\alpha_{\max}\) (\(11\) pages). From here on, the authors choose \(\alpha=\alpha_{\max}=\sqrt{N}-1\). At the end of the section a very interesting table is given (Table 1): the relation between the number of gaps and \(N\).\N\NReferences (\(3\) items).\N\NA well-written and interesting paper.