The restriction of any Freiman homomorphism of order 2 to $\{kd:-l\leq k \leq l\}$ is linear?

Question

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?

Answer 1

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 $-kd+kd=0+0$.