## A natural probability measure derived from Stern’s diatomic sequence

### Volume 183 / 2018

Acta Arithmetica 183 (2018), 87-99
MSC: Primary 11B85; Secondary 42A38, 28A80.
DOI: 10.4064/aa170709-22-1
Published online: 5 March 2018

#### 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.