# Solving a linear equation with variables with restricted prime factors

Let $a,b,c$ be pairwise coprime integers. Let $\mathcal{P}_1$, $\mathcal{P}_2$ be two sets of prime numbers with positive relative density. We can assume that the primes in each $\mathcal{P}_i$ do not divide $abc$. Let $S_i$ be the set of numbers whose prime factors all come from $\mathcal{P}_i$. Can one count the number of solutions to the equation

$$\displaystyle ax + by = c$$

with $x \in S_1, y \in S_2$ and $\max\{|x|, |y|\} < T$?

I am most interested in the case when $\mathcal{P}_i$ are primes which split (and are unramified) over quadratic fields $K_i$ for $i = 1,2$.

• I believe a sieve would give an asymptotic formula if the sum of the densities of $\mathcal P_1$ and $\mathcal P_2$ is less than $1/2$, and an upper bound of the right order of magnitude for larger densities. Presumably the answer for your most-interested case is $\asymp T^2/\log T$, but I'm not sure if one can establish an asymptotic formula. – Greg Martin May 16 '18 at 3:34