$B_h$-sets of real and complex numbers

TL;DR

The paper studies $B_h$-sets in $K=\mathbb{R}$ or $\mathbb{C}$ by associating to each $n$-tuple $\mathbf{a}$ a $B_h$-vector and its corresponding $B_h$-set. It proves that the set of $B_h$-vectors $\mathcal{B}_h$ is open and dense in $K^n$, implying that almost all $n$-element subsets are $B_h$-sets; a constructive perturbation argument using a positive separation $\Delta$ shows stability, while a partition argument with a small perturbation $\mathbf{b}$ establishes density and leads to $\mathcal{B}_\infty=\bigcap_{h\ge1}\mathcal{B}_h$ being dense via Baire’s theorem. The results extend to $B_h[g]$-sets, showing density of $\mathcal{B}_h[g]$-vectors, and it is noted that openness of these sets remains an open question. Collectively, the work demonstrates that $B_h$-type representations are robust under small perturbations in real and complex settings and lays groundwork for dense configurations of restricted sum representations.

Abstract

Let $K = \mathbb{R}$ or $\mathbb{C}$. An $n$-element subset $A$ of $K$ is a $B_h$-set if every element of $K$ has at most one representation as the sum of $h$ not necessarily distinct elements of $A$. Associated to the $B_h$ set $A = \{a_1,\ldots, a_n\}$ are the $B_h$-vectors $\mathbf{a} = (a_1,\ldots, a_n)$ in $K^n$. This paper proves that ``almost all'' $n$-element subsets of $K$ are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

Theorems & Definitions (9)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• proof