As mentioned in Terry Tao's comment to [this question](https://mathoverflow.net/q/44466/5810), it is constructively known
  

that there are primes between sufficiently large cubes. $\:$ [According to wikipedia](https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds),
  

"there exists a constant $\: \theta < 1 \:$ such that $\;\;\; \pi \hspace{-0.03 in}\left(x\hspace{-0.04 in}+\hspace{-0.04 in}x^{\hspace{.02 in}\theta}\right)-\pi \hspace{.03 in}(x) \: \sim \: \dfrac{x^{\hspace{.02 in}\theta}}{\log(x)} \;\;\;$ as $x$ tends to infinity".


Is there any somewhat-similar corresponding effective result for the density of primes in short intervals?


Motivation:  

Such a result could yield an effective randomized reduction from subset sum to [this variant of factoring](https://cstheory.stackexchange.com/q/4769/6973).



Are there effective small intervals in which primes are dense?
One can read off such an effective result from the [work of Dudek](http://arxiv.org/abs/1401.4233). Indeed, examining his Section 3.3, we can get the following explicit lower bound for $x>\exp(8\times 10^{14})$ and $h=3x^{2/3}$:
$$ \sum\_{x<p\leq x+h}\log p>h-\tfrac{1}{2}(1-10^{-3})h-\tfrac{1}{2}(1-10^{-3})h-0.0008h. $$
(Note that Dudek's definition of $E(x,h,k)$ should be multiplied by $-1$.)
That is,
$$ \sum\_{x<p\leq x+3x^{2/3}}\log p>0.0006x^{2/3},\qquad x>\exp(8\times 10^{14}).$$


Under the Riemann Hypothesis, much better explicit bounds are known, see e.g. Theorem 1.1 in the [recent preprint](http://arxiv.org/abs/1503.05403) of Dudek-Grenié-Molteni.


