Version 2 of this writeup is available, and includes a newer and simple upper bound thanks to MathOverflow 88777 as well as indirect references to future writeups. Details of further work will be found in these writeups. GRP 2014.06.04.
In a paper of Erik Westzynthius,
Here $Q$ is an abbreviation for $1$ divided by the product of the n terms $(1 - 1/p_i)$. It is roughly log(log(n)) for n for large n. Here comes the kicker. Step 4 notes that steps 1 through 3 are essentially independent of $a$, and if $x$ can be chosen so that $x/Q - 2^n > 0$, then $I_0 > 0$ which means at least one of the $q_i$ is in the interval $(a, a+x)$ when $a > 0$, and such $x$ would be an upper bound for $q_{i+1} - q_i$ which is independent of $i$. So choose $x = Q * 2^n$ plus epsilon.
