On recurrence property of Riesz-Raikov sums (Q2470017)

Let \(f\) be a real-valued function on \(\mathbb{R}\) with period 1 satisfying the following conditions: \[ \int^1_0 f(x)\,dx= 0,\quad 0<\int^1_0 f^2(x)\,dx< \infty\quad\text{and}\quad |f(x+ h)- f(x)|\leq M|h|^\alpha \] for some \(\alpha> 0\) and \(M> 0\). Among others, the Riesz-Raikov sums \(\sum^n_{k=1} f(\theta^k x)\), \(n\in\mathbb{N}\), are considered, where \(\theta> 1\). Theorem 1 states that these sums are dense in \(\mathbb{R}\) for almost every \(x\), except for the trivial case. Theorem 2 states that if \(\{\beta_k: k\in\mathbb{N}\}\subset\mathbb{R}\) is such that \(\beta_{k+1}/\beta_k\to\infty\), then the sums \(\sum^n_{k=1} f(\beta_k x)\), \(n\in\mathbb{N}\), are dense in \(\mathbb{R}\) for almost every \(x\). The proofs are sketched.