On a problem of Nathanson on minimal asymptotic bases

Abstract

Let $\mathbb{N}$ denote the set of all nonnegative integers and A be a subset of $\mathbb{N}$. Let h be an integer with $h\ge 2$. Let $n\in \mathbb{N}$ and $r_h(A,n)=♯\{(a_1,\dots ,a_h)\in A^h:a_1+\cdots +a_h=n\}$. The set A is called an asymptotic basis of order h if $r_h(A,n)\ge 1$ for all sufficiently large integer n. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. In 1988, Nathanson posed a problem on minimal asymptotic bases of order h. Recently, Chen and Tang showed that the answer to the problem is negative for $h\ge 4$ by constructing a special partition of $\mathbb{N}$. In this paper, we give a new construction of minimal asymptotic bases. This construction expands our understanding on the problem of Nathanson.

Introduction

Let $\mathbb{N}$ denote the set of all nonnegative integers and A be a subset of $\mathbb{N}$. Let h be an integer with $h\ge 2$. Let $n\in \mathbb{N}$ and

$$r_h(A,n)=♯\{(a_1,\dots ,a_h)\in A^h:a_1+\cdots +a_h=n\},$$

$$hA=\{m\in \mathbb{N}:m=a_1+\cdots +a_h,a_i\in A,i=1,2,\dots ,h\}.$$

The set A is called an asymptotic basis of order h if $r_h(A,n)\ge 1$ for all sufficiently large integer n. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. This means that for any $a\in A$, the set $E_a=hA\backslash h(A\backslash \{a\})$ is infinite.

In 1955, Stöhr [15] first introduced the concept of minimal asymptotic bases. In 1956, Härtter [6] gave a nonconstructive proof that there exist uncountably many minimal asymptotic bases of order h. In 1974, Nathanson [13] presented an explicit construction of a minimal asymptotic basis of order 2 by using binary representations. For every $h\ge 2$, Jia and Nathanson [9] gave an construction of a minimal asymptotic basis of order h. For related results concerning minimal asymptotic bases, see [3], [4], [5], [7], [8], [10], [11], [12], [16], [17].

Let W be a nonempty subset of $\mathbb{N}$. Denote by ${\mathcal{F}}^{⁎}(W)$ the set of all finite, nonempty subsets of W. Let $A(W)$ be the set of all numbers of the form $\sum _{f\in F}{2}^f$, where $F\in {\mathcal{F}}^{⁎}(W)$. Let $S_h$ be the symmetric group of h elements. In 1988, Nathanson [14] proved the following result:

Theorem A

Let $h\ge 2$. For $i=0,1,\dots ,h-1$, let $W_i=\{n\in \mathbb{N}|n\equiv i(\mathrm{mod}h)\}$. Let $A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$. Then A is a thin, strongly minimal asymptotic basis of order h.

Nathanson [14] also posed the following problem (see also Jia and Nathanson [9]):

Problem 1.1

Let $\mathbb{N}=W_0\cup W_1\cup \cdots \cup W_{h-1}$ be a partition such that $\omega \in W_i$ implies either $\omega -1\in W_i$ or $\omega +1\in W_i$. Is

$$A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$$

a minimal asymptotic basis of order h?

In 1989, Jia and Nathanson [9] obtained the following result:

Theorem B

Let $h\ge 2$ and $t=\lceil \log (h+1)/\log 2\rceil$. Partition $\mathbb{N}$ into h pairwise disjoint subsets $W_0,W_1,\dots ,W_{h-1}$ such that each set $W_i$ contains infinitely many intervals of t consecutive integers. Then $A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$ is a minimal asymptotic basis of order h.

In 2011, Chen and Chen [1] proved Theorem B under the assumption only required that each set $W_i$ contains one interval of t consecutive integers.

Theorem C

Let $h\ge 2$ and t be the least integer with $t>\log h/\log 2$. Let $\mathbb{N}=W_0\cup W_1\cup \cdots \cup W_{h-1}$ be a partition such that each set $W_i$ is infinite and contains t consecutive integers for $i=0,1,\dots ,h-1$. Then $A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$ is a minimal asymptotic basis of order h.

Recently, Chen and Tang [2] showed that the answer to the Problem 1.1 is negative for $h\ge 4$ by constructing a special partition of $\mathbb{N}$.

Theorem D

Let h and t be integers with $2\le t\le \log h/\log 2$. Then there exists a partition $\mathbb{N}=W_0\cup W_1\cup \cdots \cup W_{h-1}$ such that each set $W_i$ is a union of infinitely many intervals of at least t consecutive integers and $A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$ is not a minimal asymptotic basis of order h.

In this paper, we give a new construction of minimal asymptotic bases. This construction expands our understanding on the problem of Nathanson.

Theorem 1.1

Let h and t be positive integers with $h\ge 2$. Then there exists a partition $\mathbb{N}=W_0\cup W_1\cup \cdots \cup W_{h-1}$ such that each set $W_i$ is a union of infinitely many intervals of t consecutive integers and

$$A=A(W_0)\cup A(W_1)\cup \cdots \cup A(W_{h-1})$$

is a minimal asymptotic basis of order h.