Introducing Minkowski Normality

Abstract

We introduce the concept of Minkowski normality; a new type of normality that is related to the continued fraction expansion. Moreover, we use the ordering of rationals that is obtained from the Kepler tree to show that the sequence 1/2, 1/3 , 2/3, 1/4, 3/4, 2/5, 3/5, 1/5, ,··· can be used to give a concrete construction of an infinite continued fraction expansion of which the digits are distributed according to the Minkowski question mark measure. We define an explicit correspondence between continued fraction expansions and binary codes to show that we can use the dyadic Champernowne number to prove normality of the constructed number. Furthermore we provide a generalised construction that is based on the underlying structure of the Kepler tree. This generalisation shows that any construction that concatenates the continued fraction expansions of all rationals, ordered increasingly, based on the sum of the digits of their continued fraction expansion, results in a number that is Minkowski normal.