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 $\mathrm e(x)=\mathrm e^{2\pi i x}$, is pseudorandom in the following sense: for all $r\in\mathbb N$ the limit $$\gamma_r= \lim_{N\rightarrow\infty}\frac 1N\sum_{0\leq n<N}g(n+r)\overline{g(n)}$$ exists and we have $$\lim_{R\rightarrow\infty}\frac 1R\sum_{0\leq r<R}\bigl\lvert \gamma_r\bigr\rvert^2=0.$$