## Prime divisors of the difference set

Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few differences $a_j-a_i\ (1\le i<j\le n)$ as possible -- but at least one of them. What can be done in this direction (without assuming anything about the multiplicative structure of $N$)? Just as an example, can one guarantee that there exists a prime dividing at least one and at most $O(\sqrt N)$ of the differences $a_j-a_i$ Ideally, I'd like to have a prime dividing just $O(1)$ of the differences, but, I suspect, this is out of reach.

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 if you take $A$ to be the full set ${1,\dots,N}$, your best bet is to take the largest prime \$p