Pseudorandomness of the Ostrowski sum-of-digits function

Abstract

For an irrational $α\in(0,1)$, we investigate the Ostrowski sum-of-digits function $σ_α$. For $α$ having bounded partial quotients and $\vartheta\in\mathbb R\setminus\mathbb Z$, we prove that the function $g:n\mapsto \mathrm e(\vartheta σ_α(n))$, where $\mathrm e(x)=\mathrm e^{2πi x}$, 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 \[\lim_{R\rightarrow\infty}\frac 1R\sum_{0\leq r<R}\bigl\lvert γ_r\bigr\rvert^2=0.\]