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

## TL;DR

This paper constructs a natural probability measure from Stern's diatomic sequence, demonstrating it is purely singular continuous with a H"older continuous distribution function, and relates it to the dilation equation.

## Contribution

It introduces a new probability measure derived from Stern's diatomic sequence and analyzes its singular continuous nature and relation to the dilation equation.

## Key findings

- The measure is purely singular continuous.
- The distribution function is strictly increasing and H"older continuous.
- The measure relates to the solution of the dilation equation.

## 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, H\"older continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern's diatomic sequence.