Chu 2023 threshold best two term underapproximation egyptian

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theorem_1_12: States that for p/q with q odd and 2 the least j with p dividing q + j, the greedy m-term underapproximation is the unique best one for every m, the larger class of rationals cited for problem 206.

theorem_1_3: States that the greedy two-term underapproximation of p/q is the unique best one whenever the least j with p dividing q + j is at most 3, except for 10/17, and that this fails for some fraction at every larger value.

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Hung Viet Chu, A Threshold for the Best Two-term Underapproximation by Egyptian Fractions. Indag. Math. (N.S.) 35 (2024), no. 2, 350--375, doi:10.1016/j.indag.2024.01.006; arXiv:2306.12564 (2023). The arXiv record (https://arxiv.org/abs/2306.12564, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

For p/q with p < q, let Upsilon(p,q) be the least positive j with p dividing q + j. The paper proves that when Upsilon(p,q) <= 3 and p/q is not 10/17 the greedy algorithm G produces the unique best two-term underapproximation of p/q, that is 1/x_1 + 1/x_2 < p/q implies 1/x_1 + 1/x_2 <= 1/a_1 + 1/a_2 (Section 2; for 10/17 the greedy pair 1/2 + 1/12 is best but ties with 1/3 + 1/4), and that for each k >= 4 some p/q with Upsilon(p,q) = k has a two-term underapproximation strictly closer to p/q than the greedy one (Section 3), so the threshold 3 is sharp; the work extends Nathanson's results on two-term underapproximation. Section 4 studies stepwise underapproximation: with e_m = theta - sum_{n<=m} 1/a_n the mth error, it compares the greedy term 1/a_m against a superior underapproximation N/b_m of e_{m-1} with N >= 2, and characterizes equality 1/a_m = N/b_m, one form of the characterization being a_{m+1} >= N a_m^2 - a_m + 1. Consequently for rational theta equality holds for only finitely many m, while there exist irrational theta for which it holds for every m; Section 5 computes G(theta) for a family of theta. The tools are the greedy recursion a_1 = G(theta), a_m = G(theta - sum_{n<m} 1/a_n) with G(theta) = floor(1/theta) + 1, and the growth bounds a_1 >= 2, a_{n+1} >= a_n^2 - a_n + 1. This bears on problem 206, on how well greedy Egyptian-fraction expansions approximate a rational from below, by showing that greedy is optimal at two terms whenever Upsilon(p,q) <= 3 and that this can fail for every larger value.

Source: https://arxiv.org/abs/2306.12564.

The retained folder-name PDF is arXiv:2306.12564v2 (20 January 2024), 26 pages; v1 is of 19 June 2023; the journal version (Indag. Math. (N.S.) 35 (2024), 350--375, March 2024) has not been compared with it. Beyond the two-term threshold, Theorem 1.12 (p. 6) shows that for p < q with q odd and Upsilon(p,q) = 2 the greedy m-term sum is the unique best m-term underapproximation of p/q for every m, answering Nathanson's question whether rationals other than those with p dividing q+1 (Nathanson's Theorem 5, restated as Theorem 1.11) have this property; this is the larger class of rationals that the site's commentary on problem 206 attributes to the paper. Read status: claims checked for Theorem 1.3 and Theorem 1.12 (statements read clause by clause on PDF pp. 3 and 6), compiled on Theorem 1.3 and Theorem 1.12; the proofs were not read; nothing has been independently reviewed.

Bears on. #206

Results to transcribe.

• Theorem 1.3 (Upsilon <= 3): If Upsilon(p,q) <= 3 and p/q is not 10/17, the greedy pair 1/a_1 + 1/a_2 is the unique best two-term underapproximation of p/q; for 10/17 it is best but not unique.
• Sharpness (Upsilon >= 4): For each k >= 4 there exist p < q with Upsilon(p,q) = k for which some two-term underapproximation of p/q beats the greedy pair.
• Theorem 1.12: For p < q with q odd and Upsilon(p,q) = 2, the greedy m-term sum is the unique best m-term underapproximation of p/q for every m.
• Stepwise characterization: 1/a_m equals the superior underapproximation N/b_m of the error e_{m-1} precisely under conditions such as a_{m+1} >= N a_m^2 - a_m + 1.
• Rational versus irrational: For rational theta equality 1/a_m = N/b_m holds for only finitely many m; some irrational theta satisfy it for all m.