Compression and complexity for sumset sizes in additive number theory

TL;DR

This paper explores the structure, complexity, and compression algorithms related to sumsets of finite integer and lattice point sets, focusing on their sizes and geometric properties.

Contribution

It introduces a compression algorithm for sumsets with large diameter, enabling construction of smaller-diameter sets with identical sumset sizes.

Findings

Compression algorithm for sumsets with large diameter

Analysis of geometric and computational complexity of sumset sets

Characterization of all possible sumset sizes for integer and lattice point sets

Abstract

The study of sums of finite sets of integers has mostly concentrated on sets with small sumsets (Freiman's theorem and related work) and on sets with large sumsets (Sidon sets and $B_h$-sets). This paper considers the sets ${\mathcal R}_{\mathbf Z}(h,k)$ and ${\mathcal R}_{{\mathbf Z}^n}(h,k)$ of \emph{all} sizes of $h$-fold sums of sets of $k$ integers or of $k$ lattice points, and the geometric and computational complexity of the sets ${\mathcal R}_{\mathbf Z}(h,k)$ and ${\mathcal R}_{{\mathbf Z}^n}(h,k)$. For sumsets $hA$ with large diameter, there is a compression algorithm to construct sets $A'$ with $|hA'| = |hA|$ and small diameter.