Elsholtz 2021 sums four more unit fractions approximate

../

corollary_3: States the bound f_k(1,1) below c_0 to the power (2/5 + epsilon) 2^(k-1) for k at least k(epsilon), with c_0 = 1.5979..., for the number of k-term unit-fraction representations of 1, derived from Theorem 2's lifted bound on f_k(m,n).

theorem_1: Elsholtz and Planitzer's direct upper bound for the number f_4(m,n) of representations of m/n as a sum of four unit fractions, which with the earlier bounds gives five ranges of m in terms of n.

theorem_2: Elsholtz and Planitzer's upper bound for the number f_k(m,n) of representations of m/n as a sum of k unit fractions, k at least 5, lifted from their four-fraction bound and improving the constant in the exponent to 8/5.

---

Elsholtz, Christian and Planitzer, Stefan, Sums of four and more unit fractions and approximate parametrizations. Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709.

The copy read for this card is the fourteen-page arXiv version v1 (10 December 2020, the only arXiv version). The journal version, Bull. Lond. Math. Soc. 53 (2021), no. 3, 695--709, doi:10.1112/blms.12452, published online 25 January 2021, was not obtained or compared; the locators below are the preprint's pages and labels. Read status: Theorem 1 (p. 2), Theorem 2 (p. 4), Corollary 3 with Remark 3 (p. 5) and Conjecture 1 (p. 2) were read clause by clause on the PDF pages (claims checked); the proofs were not read beyond their structure and none has been independently reviewed. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2012.05984), every other right reserved.

The paper counts solutions f_k(m,n) of m/n = 1/a1 + ... + 1/ak with a1 at most ... at most ak, and attacks the case k = 4 directly instead of lifting from k = 3. Theorem 1 proves f_4(m,n) much less than n^eps min(n^{3/2}/m^{3/4}, n^{8/5}/m), which combined with the earlier bounds of Browning-Elsholtz and of Elsholtz-Planitzer gives five different ranges of m relative to n, improving the most relevant cases where m is small and where m is close to n. The method replaces complete parametrizations of the solution set by what the authors call approximate parametrizations: 'defining sets', sets of parameters which, once fixed, leave at most O_eps(n^eps) choices for the remaining parameters; a computer algebra system finds many defining sets and products of parameters that are small in terms of n and whose factors split into defining sets. The improvement for k = 4 propagates by the standard lifting to upper bounds on f_k(m,n) for k greater than 4. Conjecture 1, which the authors think quite possibly true, states f_k(m,n) much less than exp(C_{m,k} log n / log log n) for fixed k and m, going further than Heath-Brown's suggestion to Elsholtz that even f_3(m,n) = O_eps(n^eps) appears possible. For Problem 148, which asks for good estimates of the number F(k) of representations of 1 by k distinct unit fractions, the paper contributes the upper-bound side through its bounds on f_k(m,n) for k at least 4: Theorem 2 lifts Theorem 1 to f_k(m,n) much less than (kn)^eps (k^{4/3} n^2 / m)^{(8/5) 2^{k-5}} for k at least 5, and Corollary 3(2) gives f_k(1,1) < c_0^{(2/5+eps) 2^{k-1}} for k at least k(eps), with c_0 = lim u_n^{2^{-n}} = 1.5979... for u_0 = 1, u_{n+1} = u_n(u_n+1); since F(k) is at most f_k(1,1), this is the upper bound for F(k). The constant here is the square of the Vardi constant 1.2640..., which the site's commentary calls c_0; see the result page.

Source: https://arxiv.org/abs/2012.05984.

Bears on. #148: an upper bound only. Since the count F(k) of representations of 1 by k distinct unit fractions is at most f_k(1,1), Corollary 3(2) gives F(k) < c_0^{(2/5+eps) 2^{k-1}} for k at least k(eps), with c_0 = 1.5979.... The paper does not write out the proof of Corollary 3; it says (p. 4) that the proof of its earlier references goes through with Theorem 2's bound plugged in, and Theorem 2 rests on Theorem 1. The paper gives no lower bound for F(k).

Results.

• Theorem 1 (p. 2): For all m, n: f_4(m,n) much less than n^eps min(n^{3/2}/m^{3/4}, n^{8/5}/m), improving the earlier four-fraction bounds, displays (5) and (7) on p. 2, when m is small or close to n (Corollary 2, p. 3: m much less than n^{50/289}, or n^{4/5} much less than m).
• Theorem 2 (p. 4): For k at least 5, f_k(m,n) much less than (kn)^eps (k^{4/3} n^2 / m)^{(8/5) 2^{k-5}}, by lifting Theorem 1 (Section 5).
• Corollary 3 (p. 5): f_k(1,1) much less than k^{(2/15) 2^{k-1} + eps}; f_k(1,1) < c_0^{(2/5+eps) 2^{k-1}} for k at least k(eps) with c_0 = 1.5979...; and, for k at least k(eps), the bound c_0^{(2/5+eps) 2^k} for solutions of 1 = sum 1/a_i + 1/prod a_i.
• Conjecture 1 (p. 2): For fixed k and m, f_k(m,n) much less than exp(C_{m,k} log n / log log n) as n tends to infinity.
• Approximate parametrizations: Defining sets, sets of parameters which, once fixed, leave at most O_eps(n^eps) choices for the remaining parameters, replace full parametrizations, making a computational search feasible.

No file of this source is held. The journal version is published open access under CC BY 4.0: the license statement of its PubMed Central copy, PMC8248158, begins "This is an open access article under the terms of the http://creativecommons.org/licenses/by/4.0/ License" (record read on 2026-10-07 at https://www.ebi.ac.uk/europepmc/webservices/rest/PMC8248158/fullTextXML), and the Crossref record of doi:10.1112/blms.12452, read the same day, names the same license. That version was not obtained, and the card cites the edition it names above.