Recurrence of the integral of a smooth conditionally periodic function (Q1280637)

Let \(f\) be a function analytic in some complex neighbourhood of the real \(s\)-dimensional torus \(T^s\), \(f:T^s\to\mathbb{R}\). Let \(\omega_1,\dots,\omega_s\in\mathbb{R}\) be linearly independent over the ring \(\mathbb{Z}\) of integers and let \(\varphi=(\varphi_1,\dots,\varphi_s)\in\mathbb{R}^s\). Suppose that the spatial mean of \(f\) is zero. Denote \(I(T,\varphi)=\int^T_0f(\omega_1t+\varphi_1,\dots,\omega_st+\varphi_s)dt\). It is proved that \(I(T,\varphi)\) is recurrent for every \(\varphi\), i.e., for every \(\varepsilon>0\) and \(T>0\) there exists a \(T^*>T\) such that \(| I(T^*,\varphi)|<\varepsilon\). If, moreover, \(f(\varphi)\neq 0\), then \(I(T,\varphi)\) is oscillatory, i.e., it changes its sign infinitely many times as \(T\to\infty\).