Small quotients of smooth numbers I think this is a very interesting question, and don't believe it is known. Note that it implies your other question on logarithms of ratios of square-free numbers, which also is unknown I think. Finally, it may be worth pointing out that the kind of bound you are asking for is best possible -- random products of the first $k$ primes will cluster, and then use pigeonhole to find two near each other.

When has the Borel-Cantelli heuristic been wrong? How about the Maier phenomenon that occasionally there are intervals around $x$ of length $(\log x)^{100}$ say with substantially more (or fewer) primes than one would expect?

distribution of $\sqrt{-1} \mod p$ Hooley proved the equidistribution of roots $\mod d$ for composite $d$. That is a much easier problem than the one resolved by Duke, Friedlander and Iwaniec.