Pseudorandomness of the Ostrowski sum-of-digits function

Abstract

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

9 pages

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