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Compression and complexity for sumset sizes in additive number theory

Melvyn B. Nathanson

TL;DR

This work probes the full range of possible sizes of h-fold sumsets $|hA|$ for k-element sets $A$ within integers and lattice points. It introduces compression techniques that reduce the diameter of $A$ without changing $|hA|$, and proves an additive isomorphism $\mathcal{R}_{\mathbf{Z}^n}(h,k)=\mathcal{R}_{\mathbf{Z}}(h,k)$ via Freiman isomorphisms. The lattice-point compression is extended to directional diameters, enabling iterative diameter reduction in $\mathbf{Z}^n$. A concrete upper bound $N(h,k)<4(8h)^{k-1}$ is established for the finite-interval realization problem, using modular Freiman isomorphisms and Bertrand’s postulate. Overall, the paper blends geometric compression with Freiman theory to characterize sumset size spectra and provide quantitative bounds in additive number theory.

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.

Compression and complexity for sumset sizes in additive number theory

TL;DR

This work probes the full range of possible sizes of h-fold sumsets for k-element sets within integers and lattice points. It introduces compression techniques that reduce the diameter of without changing , and proves an additive isomorphism via Freiman isomorphisms. The lattice-point compression is extended to directional diameters, enabling iterative diameter reduction in . A concrete upper bound is established for the finite-interval realization problem, using modular Freiman isomorphisms and Bertrand’s postulate. Overall, the paper blends geometric compression with Freiman theory to characterize sumset size spectra and provide quantitative bounds in additive number theory.

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 -sets). This paper considers the sets and of \emph{all} sizes of -fold sums of sets of integers or of lattice points, and the geometric and computational complexity of the sets and . For sumsets with large diameter, there is a compression algorithm to construct sets with and small diameter.

Paper Structure

This paper contains 5 sections, 9 theorems, 176 equations.

Table of Contents

  1. Geometric and computational complexity
  2. Compression theorem for integers
  3. Additive isomorphism $\mathcal{R}_{\mathbf Z^n}(h,k) = \mathcal{R}_{\mathbf Z}(h,k)$
  4. Compression theorem for sums of sets of lattice points
  5. An upper bound for $N(h,k)$

Key Result

Theorem 1

Let $k \geq 3$ and let $A = \{a_1 < a_2 < \cdots < a_k\}$ be a set of $k$ integers. Let $h$ be a positive integer. If there exist an integer $j \in [1,k-1]$ and a positive integer $\delta_j$ such that then the integer $j$ is unique and the strictly increasing sequence of $k$ integers defined by and the set $A' = \{a'_1 < a'_2 < \cdots < a'_k\}$ satisfy Moreover, and

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 8 more