Let $B$ be a subset of $\mathbb{Z}\_N$, where $\mathbb{Z}\_N$ is the group of all congruence classes mod $N$. The function $\phi:B\rightarrow \mathbb{Z}\_N$ is said to be a ***Freiman homomorphism of order $k$*** for $k\leq2$ if
$$\phi(a\_1)+\cdots+\phi(a\_k)=\phi(a'\_1)+\cdots+\phi(a'\_k)$$
for all $a\_1,\ldots,a\_k,a'\_1,\ldots,a'\_k\in B$ such that
$$a\_1+\cdots+a\_k=a'\_1+\cdots+a'\_k.$$  

Obviously, if $\phi$ is linear, then $\phi$ is a Freiman homomorphism of any order $k$. Now, we restrict $B$ to be $\{sd:-l\leq s \leq l, d\in\mathbb{Z}\_N\}$, can we conclude that any Freiman homomorphism of order 2 on $\{sd:-l\leq s \leq l, d\in\mathbb{Z}\_N\}$ is linear?



The restriction of any Freiman homomorphism of order 2 to $\{kd:-l\leq k \leq l\}$ is linear?
Not necessarily linear (consider $\phi(x)=x+C$ for some fixed $C\in\mathbb Z\_N$), but certainly affine. To see this, let $C:=\phi(0)$ and $D:=\phi(d)-C$. We have then $\phi(d)=C+D$, $\phi(2d)=C+2D$ in view of $2d+0=d+d$ (implying $\phi(2d)+\phi(0)=\phi(d)+\phi(d)$) etc, up to $\phi(ld)=C+lD$ (following from $ld+0=(l-1)d+d$). Also, $\phi(-kd)=C-kD$ for each $k\in[1,l]$ in view of $-kd+kd=0+0$.


