For a prime $p$, let $\varphi\_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.



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> Given an integer $n\ge 3$, what is the smallest $\varepsilon=\varepsilon(n)>0$ such that for any subset $A\subset\mathbb Z$ with $|A|=n$, there exists a prime $p$ satisfying $(1-\varepsilon(n))n<|\varphi\_p(A)|<n$?
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Prime divisors of $\prod(a_i-a_j)$
Maybe I misunderstand the question, but doesn't the set $A=\{i\cdot n!\,|\,1\le i\le n\}$ have $\lvert\varphi\_p(A)\rvert=1$ for all $p$ for which $\varphi\_p$ is not injective on $A$?


