Hitting times of shrinking targets: Transversality and an ergodic theorem
Vicente Saavedra-Araya
TL;DR
This work analyzes sets Λ_y defined by hitting shrinking targets through fractional parts {u_n y}, with shrinking intervals I_n of measure n^{−a}, and establishes both fractal-transversality with arbitrary A ⊂ N and ergodic-average convergence properties. By developing precise covariance and higher-order correlation bounds for the associated indicator variables X_n(y) = 1_{I_n}({u_n y}), the authors extend LLN-type results to real, non-lacunary sequences and prove that, under δ-sublacunarity and a + δ < 1, Λ_y is a good sequence for pointwise convergence of ergodic averages in all measure-preserving systems, and that A ∩ Λ_y has controlled mass-dimension growth with explicit formulas dim_M(A ∩ Λ_y) = max{0, dim_M(A) − a} under suitable conditions. The paper further demonstrates a transversality phenomenon: for almost every y, the intersection of Λ_y with any A is dimensionally small, reflecting a geometric independence between Λ_y and A. As a consequence, the shrinking-target sets exhibit robust ergodic behavior and predictable fractal-intersection structure, with applications to sparse random-like subsequences and implications for growth patterns such as α^{n^b} y-slicing. The results synthesize fractal geometry, probabilistic dependence control, and ergodic theory to advance understanding of shrinking targets and transversality in dynamical settings.
Abstract
In this paper, we investigate ergodic and fractal properties of the sets $$Λ_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ 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 $Λ_y$ 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 $Λ_y$ and $A$ are geometrically independent for almost every $y\in[0,1]$, as witnessed by the integer-fractal dimension of their intersection