Let $A,B\subseteq \mathbb{N}$. We say that $A\sim B$ if there exists a bijection $f:A\to B$ such that:
$$\forall a_1,a_2\in A,\quad b_1,b_2\in B\quad a_1+a_2 = b_1+b_2 \iff f(a_1)+f(a_2) = f(b_1)+f(b_2)$$
This is a Frieman 2-Isomorphishm.
Prove that there exist a function $g:\mathbb{N}\to\mathbb{N}$ such that any set $A\subseteq\mathbb{N}$ with $n$ elements, has an equivalent set $B\subseteq\mathbb{N}\ \ (A\sim B)$ with a diameter (the difference between maximal and minimal element) at most $g(n)$.
Remark: the following example shows that $g(n)\geq 2^{n-2}$
Let $A_n = \{1\}\cup\{2^i+1 \mid 1\leq i< n \}$
This is true since every set $B$ that is equivalent to $A$ must satisfy:
$b_1 + b_{i+1} = 2\cdot b_i$
and therefore by setting any two elements in $B$ all of the other elements are determined.