AI 中文总结
该研究将埃及下逼近问题转化为最优控制问题，证明正有理数存在最终贪心最优埃及下逼近，还构造了具有唯一此类逼近的无理数，解答了Erdős、Graham及Nathanson的相关问题。
AI 中文摘要
我们证明，每个正有理数都存在最终贪心最优的埃及下逼近，无论分母是否允许重复，还是要求分母互不相同。这肯定地回答了源自Erdős和Graham、后由Nathanson重新研究的一个问题，并给出了关于收敛到某些有理数的单位分数序列中分母最大渐近增长的应用。我们将该问题重新表述为动力系统的最优控制问题，构造了合适的收益函数，并研究了相关Bellman函数的性质。我们还通过构造一个具有唯一且贪心最优埃及下逼近的无理数，回答了Nathanson的另一个问题。
英文摘要
We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erdős and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.
Commentsv2: 30 pages; Theorem 4, Figure 1, Remark 6, and Section 4 were added after discussions with colleagues
