<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{u_n\}_{n=1}^{\infty}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> be Sylvester’s sequence (sequence A000058 in the OEIS), and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a_1 < a_2 < \cdots$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo><</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo><</mo> <mo>⋯</mo> </mrow> </math></EquationSource> </InlineEquation> be any other sequence of positive integers satisfying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sum_{i=1}^\infty \frac{1}{a_i} = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <mfrac> <mn>1</mn> <msub> <mi>a</mi> <mi>i</mi> </msub> </mfrac> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Erdős and Graham conjectured that<Equation ID="Equa"> <EquationSource Format="TEX">$ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mo movablelimits="true">lim inf</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <msubsup> <mi>a</mi> <mi>n</mi> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mi>n</mi> </msup> </mfrac> </msubsup> <mo><</mo> <munder> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <msubsup> <mi>u</mi> <mi>n</mi> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mi>n</mi> </msup> </mfrac> </msubsup> <mo>=</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.264085</mn> <mo>…</mo> <mo>.</mo> </mrow> </math></EquationSource> </Equation> This conjecture has recently been resolved constructively by Kamio and independently by the present authors. In this paper, we focus on a generalization of this conjecture using a non-constructive approach. Specifically, assuming the unproven claim of Erdős and Graham that every rational number admits an eventually greedy best Egyptian underapproximation, we establish a more general asymptotic inequality involving best Egyptian underapproximations.</p>

Generalizing a conjecture of Erdős and Graham via best Egyptian underapproximations

期刊论文
发表日期：

2025年10月13日

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Z. Li,
Q. Tang

摘要

Let $\{u_n\}_{n=1}^{\infty}$ ${u_{n}}_{n = 1}^{\infty}$ be Sylvester’s sequence (sequence A000058 in the OEIS), and let $a_1 < a_2 < \cdots$ $a_{1} < a_{2} < \dots$ be any other sequence of positive integers satisfying $\sum_{i=1}^\infty \frac{1}{a_i} = 1$ $\sum_{i = 1}^{\infty} \frac{1}{a_{i}} = 1$ . Erdős and Graham conjectured that $ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots.$ $\underset{n \to \infty}{lim inf} a_{n}^{\frac{1}{2^{n}}} < lim_{n \to \infty} u_{n}^{\frac{1}{2^{n}}} = c_{0} = 1.264085 \dots .$ This conjecture has recently been resolved constructively by Kamio and independently by the present authors. In this paper, we focus on a generalization of this conjecture using a non-constructive approach. Specifically, assuming the unproven claim of Erdős and Graham that every rational number admits an eventually greedy best Egyptian underapproximation, we establish a more general asymptotic inequality involving best Egyptian underapproximations.