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arXiv · 2406.07218

On eventually greedy best underapproximations by Egyptian fractions

Vjekoslav Kovač
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Abstract

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero. [This note solves Problem 206 on Bloom's website "Erd\H{o}s problems".]

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Vjekoslav Kovač. 2024-06-11. On eventually greedy best underapproximations by Egyptian fractions. https://doi.org/10.1016/j.jnt.2024.09.004

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