关于埃及分数最终贪心最佳下逼近
数论
2024-11-25 v3 经典分析与常微分方程
组合数学
摘要
Erdős 和 Graham 认为，对于足够大的 n，几乎所有正数的最佳 n 项埃及下逼近都是以贪心方式构造的，即基于最佳 (n-1) 项埃及下逼近。我们证明事实恰恰相反：具有此性质的实数集合的勒贝格测度为零。[本注释解决了 Bloom 网站“Erdős 问题”上的问题 206。]
引用
@article{arxiv.2406.07218,
title = {On eventually greedy best underapproximations by Egyptian fractions},
author = {Vjekoslav Kovač},
journal= {arXiv preprint arXiv:2406.07218},
year = {2024}
}
备注
7 pages; v2 corrects the history of the claim for rational numbers; v3 incorporates referee's suggestions. The author is grateful to Richard Green for discussing the paper and its background in the popular newsletter "A Piece of the Pi: mathematics explained"