Rectifiable curves in Sierpiński carpets (Q2884651)

The authors introduce the coordinates in the carpets by NEWLINE\[NEWLINEa =(a_1^{-1}, a^{-1}_2, \cdots ) \in \biggl\{ \frac13, \frac15, \frac17,\cdots\biggr\}{}^NNEWLINE\]NEWLINE and NEWLINE\[NEWLINER_0 = Q = \biggl\{(x,y) \in \mathbb R^2 : 0 \leq x,y \leq 1\biggr\}; R_1 = \bigcup^{{a_1{}^2 -1}}_{j=1} Q_{1,j},NEWLINE\]NEWLINE with squares \(Q_{1, j}\subset Q\) of side length NEWLINENEWLINE\[NEWLINEa_1^{-1} ;\cdots; R_k =\bigcup_{j=1}^{(a{{_1^2 -1})\cdots (a{_k^2 -1})}} Q_{k,j},NEWLINE\]NEWLINENEWLINE with squares \(Q_{k , j} \subset Q_{k-1,j}\) of side length \(a^{-1}_1 \cdots a^{-1}_k\). A generalized Sierpinski carpet is defined and denoted by \(\mathbb S_a= \bigcap_{k\geq 1} R_k\). Consider the set of slopes, in \([0,1]\), of nontrivial line segments contained in carpet \(\mathbb S_a\), and denoted by Slopes \(\mathbb S_a\) .NEWLINENEWLINEThe main aims of this paper are: prove some properties of the set Slopes \(\mathbb S_a\) characterize the relationship between Slopes \(\mathbb S_a\) and Farey sequences; give a necessary condition for a line segment to lie in the carpet \(\mathbb S_a\); finally, show differentiable and rectifiable curves in the carpets.