Spectral analysis of a family of binary inflation rules

The family of primitive binary substitutions defined by $1 \mapsto 0 \mapsto 0 1^m$ with $m\in\mathbb{N}$ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation ($m=1$), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous.