On strong infinite Sidon and B_h sets and random sets of integers

2019/11/29 by David Fabian, Juanjo Rué, Fabian, David +3
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1911.13275

openalex publication_date 2019/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A set of integers S ⊂ ℕ is an α-strong Sidon set if the pairwise sums of its elements are far apart by a certain measure depending on α, more specifically if | (x+w) - (y+z) | ≥ max \ x^α,y^α,z^α,w^α\ for every x,y,z,w ∈ S satisfying max \x,w\ ≠ max \y,z\. We obtain a new lower bound for the growth of α-strong infinite Sidon sets when 0 ≤ α< 1. We also further extend that notion in a natural way by obtaining the first non-trivial bound for α-strong infinite B_h sets. In both cases, we study the implications of these bounds for the density of, respectively, the largest Sidon or B_h set contained in a random infinite subset of ℕ. Our theorems improve on previous results by Kohayakawa, Lee, Moreira and Rödl.

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