Hitting times of shrinking targets: Transversality and an ergodic theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915541012185088 |
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| author | Saavedra-Araya, Vicente |
| author_facet | Saavedra-Araya, Vicente |
| contents | 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 |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hitting times of shrinking targets: Transversality and an ergodic theorem Saavedra-Araya, Vicente Dynamical Systems Number Theory 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 |
| title | Hitting times of shrinking targets: Transversality and an ergodic theorem |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2510.07450 |