TITLE: An S-unit equation, with S an infinite and sparse set of primes.
QUESTION [6 upvotes]: Say we have three infinite sequences $\{a_i\},\{b_i\},\{c_i\}$ of natural numbers, satisfying the equations $$a_1+b_1=c_1,\dots, a_n+b_n=c_n,\dots $$
Assume further that $gcd(a_i,b_i,c_i)=1$ for each $i$ and that $(a_i,b_i,c_i)\neq (a_j,b_j,c_j)$ for all $i,j$. Now let's define $S$ as the set of primes $p$ which divide $a_ib_ic_i$ for at least one $i$. From the S-unit theorem we know that $S$ has to be infinite. 
Now the question is: Can $S$ be sparse? This can be taken to mean Dirichlet density zero, for example.
I haven't thought much about this but there are reasons to believe the answer is yes, indeed if there are infinitely many Mersenne primes $q=2^p-1$ then the equations $1+q=2^p$ give such a sparse $S$. However, I am looking for an unconditional result.

REPLY [2 votes]: An even simpler answer: take triples of the form $(2,n!-1,n!+1)$ for a sufficiently sparse sequence of values of $n$.