Pseudorandomness of the Ostrowski sum-of-digits function

TL;DR

This paper proves that the Ostrowski sum-of-digits function exhibits pseudorandom behavior for certain irrationals with bounded partial quotients, with implications for understanding digit sum functions in number theory.

Contribution

It establishes the pseudorandomness of the Ostrowski sum-of-digits function for irrationals with bounded partial quotients, a novel result in the study of digit sum functions.

Findings

The autocorrelation limits re shown to exist for all shifts.

The average squared autocorrelation tends to zero, indicating pseudorandomness.

The results apply to a class of irrationals with bounded partial quotients.

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 $\mathrm e(x)=\mathrm e^{2\pi i x}$, 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 \[\lim_{R\rightarrow\infty}\frac 1R\sum_{0\leq r<R}\bigl\lvert \gamma_r\bigr\rvert^2=0.\]