Additive structure in convex translates

Understand

Let $\mathcal{P}$ be a set of points in the plane, and $\mathcal{S}$ a strictly convex set of points.

• In this note, we show that if $\mathcal{P}$ contains many translates of $\mathcal{S}$, then these translates must come from a generalized arithmetic progression of low dimension.
• We also discuss an application to the unit distance conjecture.