Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it enables us to prove that its derivative, when it exists in a wide sense, can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k̄=5.31972, and ?′(x) exists, then ?′(x)=0. In the same way, if the same average is less than k̄=2log2Φ, where Φ is the golden ratio, then ?′(x)=∞. Finally some results are presented concerning metric properties of continued fractions and alternated dyadic expansions. © 2001 Academic Press.
- Minkowski's function; number systems; metric number theory