It is an open conjecture from a paper of Konyagin and myself that every $n$-element set of integers is Freiman-isomorphic to a subset of $[0,2^{n-2}]$. There are some counterexamples for small values of $n$, but it is believed that the conjecture is "essentially true". If so, your $4^n$ actually can be improved to $2^n$ (and in fact, to $2^{n-2}+1$). The set $\{0,1,2,4,\dotsc,2^{n-2}\}$ shows that $2^{n-2}+1$ is a sharp bound.
The bound $4^n$ is easy, but the proof I can think of is too long to present here. You can find better estimates in Chapter 20 of the monograph Structural Additive Theory by David Grynkiewicz.