On exponential Freiman dimension

TL;DR

This paper precisely determines the largest constant for sumset size bounds in relation to the exponential Freiman dimension, advancing understanding of additive combinatorics in Euclidean spaces.

Contribution

It provides an exact characterization of the largest constant in sumset inequalities for sets with a given exponential Freiman dimension.

Findings

Explicit constants for sumset bounds depending on Freiman dimension

Characterization of sets containing vertices of high-dimensional parallelepipeds

Advancement in additive combinatorics in Euclidean spaces

Abstract

The exponential Freiman dimension of a finite set $A \subset \mathbb{R}^{m}$, introduced by Green and Tao in 2006, represents the largest positive integer $d$ for which $A$ contains the vertices of a non-degenerate $d$-dimensional parallelepiped. For every $d \geq 1$, we precisely determine the largest constant $C_{d}>0$ (exponential in $d$) for which $$|A+A| \geq C_{d}|A| - O_{d}(1)$$ holds for all sets $A$ with exponential Freiman dimension $d$.