A natural probability measure derived from Stern's diatomic sequence

Abstract

Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, Hölder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.

13 pages, 3 figures; minor revision with some improvements and updates