Theorem 3.2 in http://arxiv.org/pdf/1405.2593.pdf shows that for any $x$ there are $\gg x\exp(-\sqrt{\log x})$ integers $x_0 \in [x; 2x]$ such that $\pi(x_0 + \log x) - \pi(x_o) \gg \log\log x$.

Is there an upper bound for number of such $x_0$? I think it must be $<x(\log x)^{-c}$ for any $C$.

Theorem 3.2 in http://arxiv.org/pdf/1405.2593.pdf shows that for any $x$ there are $\gg x\exp(-\sqrt{\log x})$ integers $x_0 \in [x; 2x]$ such that $\pi(x_0 + \log x) - \pi(x_0) \gg \log\log x$.

Is there an upper bound for number of such $x_0$? I think it must be $<x(\log x)^{-c}$ for any $C$.

UPD: It is interesting to find such upper bound for prime numbers $x_0$ such that $\pi(x_0 + \log x) - \pi(x_0) \gg \log\log x$