Hitting times of shrinking targets: transversality and an ergodic theorem

Abstract In this paper, we investigate ergodic and fractal properties of the sets $$ \begin{align*} &\Lambda_y:=\Big\{n\in{\mathbb N}:\ \{u_ny\}\in I_n\Big\},\end{align*} $$ 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.