2026/07/30 by Vjekoslav Kovač, Quanyu Tang · 1 voice
Mathematics · #math.NT #math.CA #math.CO #math.OC
26 pages
arxiv created 2026/07/30 · arxiv updated 2026/07/31
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 certain rational numbers. 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.