Nathanson 2023 underapproximation egyptian fractions

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open_problem_4: Records Nathanson's formulation of the Erdős–Graham claims that every rational is eventually greedy and that some irrationals are not, asking for a proof or disproof; both have since been addressed.

theorem_5: States that for p/q with p dividing q + 1 the greedy n-term Egyptian underapproximation sequence is the unique best one for every n, the Nathanson case cited for problem 206.

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Melvyn B. Nathanson, Underapproximation by Egyptian fractions. J. Number Theory 242 (2023), 208-234, doi:10.1016/j.jnt.2022.07.005; arXiv:2202.00191 (2022). The arXiv record (https://arxiv.org/abs/2202.00191, read 2026-10-07) names the Creative Commons Attribution 4.0 license.

A nondecreasing sequence 2 <= x_1 <= ... <= x_n of integers (repeated denominators allowed) is an n-term Egyptian underapproximation sequence of theta in (0,1] if the sum of the reciprocals is less than theta; the greedy algorithm picks each term as large a reciprocal as still fits. Theorem 1 computes the greedy sequence for theta = p/q with p dividing q+1, showing a_1 = (q+1)/p and a_{k+1} = q a_1 ... a_k + 1, with remainder 1/(q a_1 ... a_k), a generalization of Sylvester's sequence, which Corollary 1 recovers as the case theta = 1. Theorem 2 gives a criterion: a sequence with a_1 >= 2 and a_{i+1} >= a_i^2 - a_i + 1 is the n-term greedy sequence of theta exactly when theta lies in an explicit interval determined by the partial reciprocal sums, with Corollaries 2 and 3 specializing to two terms and to infinite series. Theorem 3 shows the best n-term underapproximation u_n(theta) is always attained and hence rational, and Theorem 5 follows Soundararajan's method, through the Muirhead-type inequality of Theorem 4, to show that for p/q with p dividing q+1 the greedy underapproximations are the unique best ones at every length, an infinite set of rationals, while Theorems 6 and 7 list the numbers in (1/3, 1] whose two-term greedy pair fails to be best or uniquely best; Section 8 poses open problems, among them (4), the Erdős–Graham assertions on eventual greediness. For problem 206 the paper supplies the underapproximation counterpart of greedy Egyptian-fraction questions, the infinite family p/q <= 1 with p dividing q+1 where greed is uniquely best at every length (Theorem 5), and a two-term classification on (1/3, 1] (Theorems 6 and 7), both in the convention allowing repeated denominators, which problem 206 does not.

Source: https://arxiv.org/abs/2202.00191.

The retained folder-name PDF is arXiv:2202.00191v2 (3 February 2022), 20 pages; v1 is of 1 February 2022; the journal version (J. Number Theory 242 (2023), 208--234, January 2023) has not been compared with it. Read status: claims checked for Theorem 1, Theorem 5 and Open problem (4) (statements read clause by clause on PDF pp. 3, 9 and 18), compiled on Theorem 5 and Open problem (4); the proof of Theorem 5 was read for structure and the other proofs were not read; nothing has been independently reviewed.

Bears on. #206

Results to transcribe.

• Theorem 1: For theta = p/q with p dividing q+1, the greedy underapproximation sequence has a_1 = (q+1)/p and a_{k+1} = q prod_{i<=k} a_i + 1, generalizing Sylvester's sequence.
• Corollary 1: For theta = 1 the infinite greedy sequence is Sylvester's sequence.
• Theorem 2: A sequence with a_1 >= 2 and a_{i+1} >= a_i^2 - a_i + 1 is the n-term greedy underapproximation sequence of theta iff theta lies in an explicit interval of partial reciprocal sums.
• Theorem 3: For every theta in (0,1] and every n the best n-term Egyptian underapproximation u_n(theta) is attained by some admissible sequence and is therefore rational.
• Theorem 5: For theta = p/q with p dividing q+1 and every n, the greedy n-term sequence is the unique n-term Egyptian underapproximation sequence whose sum is at least the greedy sum, so greedy is the unique best at every length.
• Open problem (4): Prove or disprove the Erdős–Graham assertions that every rational theta is eventually greedy (u_n(theta) = u_{n_0}(theta) + u_{n-n_0}(theta - u_{n_0}(theta)) for n > n_0, the remainder handled greedily) and that some irrationals are not.