# Additive structure in convex translates

## TL;DR

This paper investigates the structure of point sets containing many translates of a convex set, revealing they form low-dimensional generalized arithmetic progressions, with implications for the unit distance conjecture.

## Contribution

It establishes a structural characterization of point sets with many convex translates, linking geometric configurations to algebraic progressions.

## Key findings

- Point sets with many convex translates are contained in low-dimensional generalized arithmetic progressions.
- The results have applications to the unit distance conjecture.
- Provides a new perspective on the additive structure of geometric configurations.

## Abstract

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.