Pseudorandomness of the Ostrowski sum-of-digits function

For an irrational $\alpha \in (0,1)$, we investigate the Ostrowski sum-of-digits function ${\sigma }_{\alpha }$. For $\alpha$ having bounded partial quotients and $\vartheta \in \mathbb{R}\setminus \mathbb{Z}$, we prove that the function $g:n\mapsto \mathrm{e}(\vartheta {\sigma }_{\alpha }(n))$, where $\mathrm e(x)=\mathrm e^{2\pi i x}$,ispseudorandominthefollowingsense:forall$r\in\mathbb N$ the limit $${\gamma }_r=\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{0\le n<N}g(n+r)\overline{g(n)}$$ exists and we have $$\underset{R\to \mathrm{\infty }}{\lim }\frac{1}{R}\sum _{0\le r<R}\mid {\gamma }_r{\mid }^2=0.$$