Constructing Thick $B_h$-sets

TL;DR

This paper generalizes constructions of $B_h$-sets in commutative semigroups, providing bounds on their diameter in integers for small sizes, and discusses open problems in the area.

Contribution

It extends classical constructions of Bose-Chowla and Singer to $B_h$-sets, deriving new bounds and highlighting open questions.

Findings

Generalized Bose-Chowla and Singer constructions for $B_h$-sets

Derived bounds on the diameter of small $B_h$-sets in integers

Presented open problems for future research

Abstract

A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (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.