Liminf-results for sums with Kronecker sequence

TL;DR

This paper investigates the liminf behavior of sums involving irrational rotations and periodic functions, showing that the property of having a zero liminf is not exclusive to absolutely continuous functions and exploring related properties.

Contribution

It demonstrates that the zero liminf property for sums over irrational rotations is not unique to absolutely continuous functions and provides new results on liminf properties of these sums.

Findings

Zero liminf property is not exclusive to absolutely continuous functions.

Established new results on liminf behavior of sums with irrational rotations.

Showed that absolute continuity is not a necessary condition for zero liminf.

Abstract

For irrational $\theta$ and 1-periodic function $f$ we consider sums $\sum_0^{Q-1}f(k\theta+\varphi)$ where $\varphi \in \mathbb R$. Sidorov proved that if $f$ is absolutely continuous function, then $\liminf_{Q \to \infty} |\sum_0^{Q-1}f(k\theta+\varphi)| = 0$ for any irrational $\theta$ and any $\varphi \in \mathbb R$. The article shows that this property is not a criterion of absolute continuity, and also obtains some other results concerning the liminf-properties of these sums.