The recurrence time for irrational rotations

Abstract

Let $T$ be a measure preserving transformation on $X \subset \mathbb{R}^d$ with a Borel measure $\mu$ and $R_E$ be the first return time to a subset $E$. If $(X,\mu)$ has positive pointwise dimension for almost every $x$, then for almost every $x$ \[ \limsup_{r \to 0^+} \frac{\log R_{B(x,r)}(x)}{-\log \mu(B(x,r))} \le 1, \] where $B(x,r)$ the the ball centered at $x$ with radius $r$. But the above property does not hold for the neighborhood of the `skewed' ball. Let $B(x,r;s) = (x - r^s, x + r)$ be an interval for $s >0$. For arbitrary $\alpha \ge 1$ and $\beta \ge 1$, there are uncountably many irrational numbers whose rotation satisfy that \[ \limsup_{r \to 0^+} \frac{\log R_{B(x,r;s)}(x)}{-\log \mu (B(x,r;s))} = \alpha \quad \text{and}\quad \liminf_{r \to 0^+} \frac{\log R_{B(x,r;s)}(x)}{-\log \mu (B(x,r;s))} = \frac{1}{\beta} \] for some $s$.