On the law of logarithm of the recurrence time

Let $T$ be a transformation from $I=[0,1)$ onto itself andlet $Q_n(x)$ be the subinterval $[i/2^n,(i+1)/2^n)$, $0 \leq i $\lim_{n\to\infty}\frac{\log K_n(x)}{n}=1 \quad$and$\quad\lim_{n\to\infty}\frac{\log K_n(x,y)}{n}=1 \quad $a.e.

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