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 $s$.