Hitting times of shrinking targets: Transversality and an ergodic theorem

In this paper, we investigate ergodic and fractal properties of the sets $$\Lambda_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence of real numbers, $y\in [0,1]$ and each $I_n$ is a finite union of intervals with decreasing Lebesgue measure. Our main result shows that, under suitable conditions, the set $\Lambda_y$ is good for pointwise convergence of ergodic averages for Lebesgue almost every $y\in [0,1]$. Furthermore, we prove a transversality phenomenon: for any fixed set $A\subseteq \mathbb{N}$, the sets $\Lambda_y$ and $A$ are geometrically independent for almost every $y\in[0,1]$, as witnessed by the integer-fractal dimension of their intersection