Multiplication Tables for Integers with Restricted Prime Factors

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the case $\delta = 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $\delta = 1/\log 4$, for which we explicitly determine the behaviour.