On the system $f(nx)$ and probabilistic number theory

TL;DR

This paper surveys recent results on the asymptotic behavior of the system {f(nx)} where f is a periodic measurable function, highlighting its unique probabilistic and number theoretic properties distinct from classical harmonic analysis.

Contribution

It provides new asymptotic results for the system {f(nx)} when f is not square integrable, expanding understanding of these series in probabilistic number theory.

Findings

Asymptotic properties differ from classical harmonic analysis

Results apply to functions not square integrable

Highlights probabilistic and number theoretic effects

Abstract

Let $f: {\mathbb R}\to {\mathbb R}$ be a measurable function satisfying \begin{equation*} f(x+1)=f(x), \qquad \int_0^1 f(x)\, dx=0, \qquad \int_0^1 f^2(x)\, dx<\infty. \end{equation*} The asymptotic properties of series $\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis. In this paper we survey some recent results in the field and prove asymptotic results for the system $\{f(nx), n\ge 1\}$ in the case when the function $f$ is not square integrable.