On a problem of Nathanson related to minimal asymptotic bases
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Let $\mathbb{N}$ be the set of all nonnegative integers. Let $W$ be a nonempty subset of $\mathbb{N}$. Denote by $F^{*}(W)$ the set of all finite, nonempty subsets of $W$ and by $A(W)$ the set of all numbers of the form $\sum_{f\in F} 2^{f}$, where $F\in F^{*}(W)$. Let $\mathbb{N}=W_{0}\cup W_{1}\cup W_{2}$ be a partition such that each set $W_{i} (i=0, 1, 2)$ is infinite. In this paper, under the assumption that $W_{0}$ and $W_{1}$ contain consecutive integers and $W_{2}$ contains no consecutive integers, we show that $A=A(W_{0})\cup A(W_{1})\cup A(W_{2})$ is a minimal asymptotic basis of order three if and only if $W_{2}-1\not\subseteq W_{0}$ and $W_{2}-1\not\subseteq W_{1}$. We partially resolved the problem on minimal asymptotic bases of order three which had been posed by Nathanson.
Abstract
Let $\mathbb{N}$ be the set of all nonnegative integers. Let $W$ be a nonempty subset of $\mathbb{N}$. Denote by $F^{*}(W)$ the set of all finite, nonempty subsets of $W$ and by $A(W)$ the set of all numbers of the form $\sum_{f\in F} 2^{f}$, where $F\in F^{*}(W)$. Let $\mathbb{N}=W_{0}\cup W_{1}\cup W_{2}$ be a partition such that each set $W_{i} (i=0, 1, 2)$ is infinite. In this paper, under the assumption that $W_{0}$ and $W_{1}$ contain consecutive integers and $W_{2}$ contains no consecutive integers, we show that $A=A(W_{0})\cup A(W_{1})\cup A(W_{2})$ is a minimal asymptotic basis of order three if and only if $W_{2}-1\not\subseteq W_{0}$ and $W_{2}-1\not\subseteq W_{1}$. We partially resolved the problem on minimal asymptotic bases of order three which had been posed by Nathanson.