Construction of an exotic measure: dyadic doubling and triadic doubling does not imply doubling (Q2633852)

The authors construct in a very sophisticated way a measure \(\mu\) on \([0,1]\) which is both dyadic doubling and triadic doubling but not doubling. Recall that \(\mu\) is doubling whenever there exists \(C>0\) such that \(\mu(\tilde I)\le C \mu(I)\) for any interval \(I\) where \(\tilde I\) stands for the interval with the same midpoint and twice the length of \(I\) and \(\mu\) is called dyadic or triadic doubling when the property only holds between \(D\) and \(\tilde D\) meaning a dyadic (or triadic) interval \(D\) and its dyadic (or triadic) parent \(\tilde D\). The measure constructed coincides with the Lebesgue measure for most of the interval \([0,1]\), except a collection of dyadic intervals on which they associate certain weight function, and the construction makes use of some number-theoretic properties of dyadic and triadic rational numbers.