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${\mathbf Q}$-independence and the construction of $B_h$-sets of integers and lattice points

Melvyn B. Nathanson

TL;DR

This work addresses constructing finite $B_h$-sets in $\mathbb{Z}$ and lattice points by a simple $\mathbf{Q}$-vector-space framework. It introduces $\mathbf Q$-independent vectors $\vec{\theta}_i$ and rational-approximation lattices $A_{h,n}(q,m)$ to guarantee unique $h$-fold representations, obtaining $B_h$-sets in $\mathbb{Z}^d$ and, when $d=1$, finite $B_h$-sets of integers. A key result is that for $q>\frac{2hm}{\varepsilon_{h,n}}$, the sets $A_{h,n}(q,m)$ are $B_h$-sets with controlled growth $\|A_{h,n}(q,m)\|_{\infty} \le q\|\Theta_n\|_{\infty}+m$, and the paper provides explicit numerical instances and a $g$-adic expansion construction (yielding Sidon sets) along with open questions about the relationship between $\mathbf Q$-independence and the $B_h$-property. The approach contributes concrete, explicit finite Sidon-type sets in both integers and lattice points, enriching the classical theory of $B_h$-sets.

Abstract

This paper gives a simple ${\mathbf Q}$-vector space construction of finite $B_h$-sets of integers and lattice points.

${\mathbf Q}$-independence and the construction of $B_h$-sets of integers and lattice points

TL;DR

This work addresses constructing finite -sets in and lattice points by a simple -vector-space framework. It introduces -independent vectors and rational-approximation lattices to guarantee unique -fold representations, obtaining -sets in and, when , finite -sets of integers. A key result is that for , the sets are -sets with controlled growth , and the paper provides explicit numerical instances and a -adic expansion construction (yielding Sidon sets) along with open questions about the relationship between -independence and the -property. The approach contributes concrete, explicit finite Sidon-type sets in both integers and lattice points, enriching the classical theory of -sets.

Abstract

This paper gives a simple -vector space construction of finite -sets of integers and lattice points.

Paper Structure

This paper contains 4 sections, 3 theorems, 64 equations.

Table of Contents

  1. $B_h$-subsets of a group
  2. The set $\mathcal{X}_{h,n}$
  3. $B_h$-sets of lattice points
  4. Examples of the construction of Sidon sets

Key Result

Theorem 1

Let $\Theta_n = \{\vec{\theta}_1, \vec{\theta}_2, \ldots, \vec{\theta}_n \}$ be a set of $\mathbf Q$-independent vectors in $\mathbf R^d$. Let $h \geq 2$ and define $\varepsilon_{h,n}$ by Sidon:varepsilon. For all positive integers $q$ and $m$ with the $(2m)^{dn}$ sets $A_{h,n}(q,m)$ constructed from $\Theta_n$ are $B_h$-sets of lattice points in $\mathbf Z^d$ with

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof