Is there any $\alpha>0$ for which there are known to exist two sequences of primes, $(p_i), (q_i)$ such that
$$\alpha = \lim_{i\to\infty} \left(p_i/\ln p_i - q_i /\ln q_i\right)\ ?$$
The motivation here is to ask the simplest possible question about patterns in the primes *after* normalizing to "correct" for the prime number theorem.

More generally, one could do something like this. First look at the point set $P=\{ p/ \ln p\}$ where $p$ runs over all the primes. Then view this point set as an atomic measure on ${\Bbb R}$. Equip some nice set of measures on ${\Bbb R}$ with some appropriately weak topology and construct a dynamical flow bytaking the closure of all the left shifts of $P$.

Is anything non-trivial known about this flow?