This research demonstrates constructing subsets of unit fractions summing to one from larger subsets, indicating new potential insights into number theory.
We prove that a subset A⊆ [1, N] with align* &∑n∈ A1/n ≥ (log N)4/5 + o(1)align* contains a subset B such that align* &∑n∈ B 1/n = 1.align* Our techniques refine those of Croot and of Bloom. Using our refinements, we additionally consider a number of questions regarding unit fractions due to Erdős and Graham.
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Liu et al. (2025) studied this question.
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