= Sharpness of the linear growth bound for integrable observables
For every $0<a<1$, choose $1<p<1/a$. On the <Bernoulli shift> over independent uniform coordinates in $(0,1)$, the observable $f(x)=x_0^{-1/p}$ is integrable, but $\mu(f(T^nx)>n^a)=n^{-ap}$. The events are independent and their probabilities have divergent sum, so the <Borel-Cantelli lemma> implies that $f(T^nx)/n^a>1$ infinitely often <almost surely>. No exponent below one gives a universal bound for all integrable observables.
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