埃及分数逼近与弱贪婪算法

Pub Date : 2023-06-03 DOI:10.1016/j.indag.2023.05.008

Hùng Việt Chu

引用次数: 1

摘要

让0 & lt;θ⩽1。如果∑n=1∞1/bn=θ，则正整数序列(bn)n=1∞称为θ的弱贪心逼近。我们引入弱贪婪近似算法(WGAA)，对于每个θ，产生两个正整数序列(an)和(bn)，使得(a)∑n=1∞1/bn=θ;(b) 1/an+1<θ−∑i=1n1/bi<1/(an+1 - 1)对于所有n个小于或等于1的人;(c)存在t个小于或等于1的人，使得bn/an≤t无限频繁。然后，我们研究了WGAA何时可以产生给定的弱贪婪近似(bn)。此外，我们表明，对于具有a1小于2和an→∞的任何非递减(an)，存在θ和(bn)，使得(a)和(b)得到满足;是否满足(c)也取决于序列(an)。最后，我们讨论了θ和(bn)的唯一性，并将我们的框架应用于特定的序列。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Approximation by Egyptian fractions and the weak greedy algorithm

Let $0 < θ ⩽ 1$ . A sequence of positive integers ${(b_{n})}_{n = 1}^{\infty}$ is called a weak greedy approximation of $θ$ if $\sum_{n = 1}^{\infty} 1 / b_{n} = θ$ . We introduce the weak greedy approximation algorithm (WGAA), which, for each $θ$ , produces two sequences of positive integers $(a_{n})$ and $(b_{n})$ such that

(a) $\sum_{n = 1}^{\infty} 1 / b_{n} = θ$ ;

(b) $1 / a_{n + 1} < θ - \sum_{i = 1}^{n} 1 / b_{i} < 1 / (a_{n + 1} - 1)$ for all $n ⩾ 1$ ;

We then investigate when a given weak greedy approximation $(b_{n})$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $(a_{n})$ with $a_{1} ⩾ 2$ and $a_{n} \to \infty$ , there exist $θ$ and $(b_{n})$ such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence $(a_{n})$ . Finally, we address the uniqueness of $θ$ and $(b_{n})$ and apply our framework to specific sequences.

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