Spectrum of image of a Freiman Homomorphism

Question

Let $A\subset \mathbb{F}_p^n$ along with a k-Freiman isomorphism $\phi:A \rightarrow \mathbb{F}_p^m$ for $A$. Can we say any meaningful statement about the structure of $Spec_\alpha(\phi(A))$ comparing to $Spec_\alpha(A)$ ?

Answer 1

Freiman $k$-isomorphism preserves the number of additive $2k$-tuples. Hence, if $A\subset G$ is $k$-isomorphic to $B\subset H$, then $$ \sum_\chi |\hat1_A(\chi)|^{2k} = \sum_\psi |\hat1_B(\psi)|^{2k}, $$ where $\chi$ runs over all characters of $G$, and $\psi$ runs over all characters of $H$. This identity relates the two spectra in a somewhat indirect way, and my guess is that nothing can be said in general beyond this.