Constructing Thick B_h-sets

2023/08/23 by O'Bryant, Kevin
#11P70 (Primary) #11T99 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2308.12406

Abstract

A subset A of a commutative semigroup X is called a B_h set in X if the only solutions to a₁+…+a_h = b₁ + ⋯ +b_h (with a_i,b_i ∈ A) are the trivial solutions \a₁,…,a_h\ = \b₁,…,b_h\ (as multisets). With h=2 and X=\mathbb Z, these sets are also known as Sidon sets, Golomb Rulers, and Babcock sets. In this work, we generalize constructions of Bose-Chowla and Singer and give the resultant bounds on the diameter of a k element B_h set in \mathbb Z for small k. We conclude with a list of open problems.

Related

• On the Diameter of Finite Sidon Sets
• On the Thickness of Infinite Generalized Sidon Sets, II
• New constructions and bounds for nonabelian Sidon sets with applications to Turán-type problems
• Finding large additive and multiplicative Sidon sets in sets of integers
• B_h-sets of real and complex numbers