Discrete Fourier restriction theorems in two dimensions

Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({\mathbb{R}}_{\mathrm{d}}^2)$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({\mathbb{R}}_{\mathrm{d}}^2)$ to the boundary of a strictly convex domain when those functions vanish outside that boundary. We give visual proofs of that statement and a complementary one.