Liminf-results for sums with Kronecker sequence
Artem Chebotarenko
TL;DR
The paper studies the liminf behavior of sums $S_Q(\varphi)=\sum_{k=0}^{Q-1} f(k\theta+\varphi)$ for irrational $\theta$ and 1-periodic $f$ with zero mean, revisiting known results such as Sidorov's: for absolutely continuous $f$, $\liminf_{Q\to\infty}\max_{\varphi}|S_Q(\varphi)|=0$. It proves Theorem 1, showing there exists a continuous $f$ not of bounded variation for which $\liminf_{Q\to\infty}|S_Q(\varphi)|=0$ for all $\theta$ and $\varphi$, thereby decoupling the liminf property from absolute continuity. It also proves Theorem 2: for $\theta$ with very good rational approximations (via continued fractions), there exists a BV function $g_\theta$ with zero mean such that $\liminf_{\nu\to\infty}\sum_{k=0}^{Q_\nu-1} g_\theta(k\theta)>0$, i.e., positive liminf in the BV setting under strong arithmetical conditions. Additional propositions give converse-type results and special cases, refining how the arithmetics of $\theta$ interact with regularity assumptions on $f$ to determine liminf behavior of Kronecker-sum sequences.
Abstract
For irrational $θ$ and 1-periodic function $f$ we consider sums $\sum_0^{Q-1}f(kθ+\varphi)$ where $\varphi \in \mathbb R$. Sidorov proved that if $f$ is absolutely continuous function, then $\liminf_{Q \to \infty} |\sum_0^{Q-1}f(kθ+\varphi)| = 0$ for any irrational $θ$ 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.