The recurrence time for irrational rotations

Let T be a measure preserving transformation on X ⊂ ℝ^d with a Borel measure μ and R_E be the first return time to a subset E. If (X, μ) has positive pointwise dimension for almost every x, then for almost every x lim_r→0+ sup log R _B(x,r)(x)/-log μ(B(x, r)) ≤ 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 α ≥ 1 and β ≥ 1, there are uncountably many irrational numbers whose rotation satisfy that lim_r→0+ sup log R_B(x,r;s)(x)/-log μ(B(x, r; s)) = α and lim_r→0+ inf log R_B(x,r;s)(x)/-log μ(B(x, r; s)) = 1/β for some s.