Inversor of digits \(Q^\ast_2\)-representative of numbers (Q2028185)

The authors study the inversor \(I:[0,1]\to[0,1]\) of digits of some binary representations \(b\) of real numbers. To be precise \(I(b(s_{1}s_{2}\dots))=b(|1-s_{1}||1-s_{2}|\dots)\), where \(b\) is a binary polybasic non-self-similar representation of real numbers in \([0,1]\). They show that \(I\) continuous, strictly decreasing and typically a singular function in the sense of Lebesgue. Moreover the affinity dimension of the graph of \(I\) and the integral of \(I\) are explicitly calculated. We think that the functions studied here are interesting. Especially we wonder if the affinity dimension coincides with the Hausdorff dimension of the graphs in this context.