On Nathanson's Triangular Number Phenomenon

2025/06/25 by Kevin O’Bryant, O'Bryant, Kevin · 3 citations
Mathematics · Physics and Astronomy · #11B05 #11B13 #11B30 #11B34 #11B75 #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.20836

openalex publication_date 2025/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite set A⊆ ℤ, the h-fold sumset is hA :=\x₁+…+x_h:x_i∈ A\. We interpret the beginning of the sequence of sumset sizes (|hA|)_h=1^∞ in terms of the successive L¹-minima of a lattice (specifically, the points in ℤ^|A| whose coordinates sum to 0 and which are perpendicular to ⟨ a₁,…,a_|A|⟩). In particular, if h₁,h₂ are the first and second minima, and 1≤ h

Citations

• Triangular and tetrahedral number differences of sumset sizes in additive number theory
• Compression and complexity for sumset sizes in additive number theory
• INTERSECTIONS OF SUMSETS IN ADDITIVE NUMBER THEORY
• Visualizing the Sum-Product Conjecture

Cited by

• Triangular gaps in the most frequent sizes of hA for |A|=4
• Additive sumset sizes with tetrahedral differences
• On the size of h-fold sumsets
• Explicit sumset sizes in additive number theory

Related

• A problem on sumset sizes of sets of lattice points
• Improved stability for the size and structure of sumsets
• Triangular gaps in the most frequent sizes of hA for |A|=4
• Explicit sumset sizes in additive number theory
• Effective results on the size and structure of sumsets