Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">N</mi> </math></EquationSource> </InlineEquation> be the set of all nonnegative integers. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A\subseteq \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_1(A,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of solutions of the equation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n=a+a'$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a,a'\in A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R_2(A,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$R_3(A,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the numbers of solutions with the additional restrictions <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a<a'$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo><</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a\le a'$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≤</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, respectively. A set <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$A\subseteq \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊆</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> is called an asymptotic basis if every sufficiently large integer can be written as the sum of two elements of A. An asymptotic basis A is minimal if no proper subset of A is an asymptotic basis. In this paper, we prove that for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$i\in \{1,2,3\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, there does not exist a minimal asymptotic basis A such that <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$R_i(A,n) = R_i(\mathbb {N}\setminus A,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">N</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all sufficiently large integers n.

A note on minimal asymptotic bases

期刊论文
发表日期：

2026年4月16日

查看全文

Shi-Qiang Chen

摘要

Let $\mathbb {N}$ $N$ be the set of all nonnegative integers. For $A\subseteq \mathbb {N}$ $A \subseteq N$ and $n\in \mathbb {N}$ $n \in N$ , let $R_1(A,n)$ $R_{1} (A, n)$ denote the number of solutions of the equation $n=a+a'$ $n = a + a^{'}$ with $a,a'\in A$ $a, a^{'} \in A$ , and let $R_2(A,n)$ $R_{2} (A, n)$ and $R_3(A,n)$ $R_{3} (A, n)$ be the numbers of solutions with the additional restrictions $a<a'$ $a < a^{'}$ and $a\le a'$ $a \leq a^{'}$ , respectively. A set $A\subseteq \mathbb {N}$ $A \subseteq N$ is called an asymptotic basis if every sufficiently large integer can be written as the sum of two elements of A. An asymptotic basis A is minimal if no proper subset of A is an asymptotic basis. In this paper, we prove that for $i\in \{1,2,3\}$ $i \in {1, 2, 3}$ , there does not exist a minimal asymptotic basis A such that $R_i(A,n) = R_i(\mathbb {N}\setminus A,n)$ $R_{i} (A, n) = R_{i} (N \ A, n)$ for all sufficiently large integers n.