On the set of numbers representable as continued fractions with bounded partial quotients (Q2780496)

Let \(F(k)=\{[0,a_1,a_2,\dots]|a_i\leq k\) \((i=1,2,\dots)\}\) for a positive integer \(k\), where \([a_0,a_1,a_2,\dots]\) is the continued fraction. It is known that the set \(F(k)\) can be obtained by deleting countably many disjoint intervals from the unit interval \(\mathcal I\). That is, \(F(k)=\mathcal I\backslash\cup_i\Delta_i\). At each step an interval \(\Delta_i\) is deleted from some closed interval \(M_i\subset\mathcal I\) by partitioning it as \(M_i=N_i^1\sqcup\Delta_i\sqcup N_i^2\) and \(\min\{|N_1|,|N_2|\}\geq\tau|\Delta_i|\).NEWLINENEWLINENEWLINEThis paper describes the properties for \(\Delta_i\), \(|\Delta_i|/|N_i^1|\) (\(k=1,2,\dots,k-1\)) and \(\min_M|N_{k-1}^1|/|\Delta_{k-1}|\), where \(M\) is some closed interval, by considering the set of numbers for which the first \(r\) partial quotients are bounded by \(k\).NEWLINENEWLINENEWLINESimilar results were obtained by \textit{S. Astels} in [Trans. Am. Math. Soc. 352, 133-170 (2000; Zbl 0967.11026)].