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Every positive rational has eventually greedy best Egyptian underapproximations, with or without repeated denominators.

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2026-07-31 09:00 UTC pith:HX437XAU

load-bearing objection Resolves the Erdős–Graham eventual-greediness problem for all positive rationals via a clean Bellman/payoff argument; the hinges check out.

arxiv 2607.28387 v2 pith:HX437XAU submitted 2026-07-30 math.NT math.CAmath.COmath.OC

Eventually greedy best Egyptian underapproximations of rational numbers via optimal control

Vjekoslav Kova\v{c} , Quanyu Tang This is my paper
classification math.NT math.CAmath.COmath.OC MSC 11D6849N90
keywords Egyptian fractionsunit fractionsgreedy algorithmbest underapproximationsoptimal controlBellman functionSylvester sequence
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Egyptian underapproximations are finite sums of unit fractions kept strictly below a target number λ. The best n-term sum is the largest such sum; the greedy rule always appends the smallest legal next denominator that stays under λ. For a general rational these two notions can disagree for small n. This paper proves that the disagreement is only temporary: after some finite length n0 that depends on λ, every best (n0+m)-term underapproximation is obtained by taking a best n0-term sum and then following the greedy rule for the remaining m terms. The result holds both when equal denominators are allowed and when they must be distinct. The same argument yields a comparison of growth rates for infinite unit-fraction series that sum to certain rationals, and supplies an explicit irrational that is uniquely greedy at every length.

Core claim

For every positive rational λ and both denominator conventions (non-decreasing or strictly increasing), there exists a finite n0 such that the best (n0+m)-term Egyptian underapproximation is exactly the best n0-term underapproximation plus the unique greedy m-term underapproximation of the residual. Moreover a single maximizing n0-tuple extends by successive greedy tails to maximizing tuples of every larger length.

What carries the argument

A Bellman function B for a controlled dynamical system whose state records the current rational remainder and the last used denominator. The payoff is a specially constructed function H satisfying H(Φ(t))=2H(t) and log t ≤ H(t) ≤ log t + log 2, where Φ(t)=t(t+1) is the reciprocal map along unit-fraction tails; discounted values of terminal decompositions are then compared via dynamic-programming identities and a completion lemma.

Load-bearing premise

The argument needs a strictly increasing payoff H that exactly doubles under the unit-fraction map and grows only logarithmically, so that any better-than-greedy long competitor can be completed to a terminal decomposition whose discounted value exceeds the optimal Bellman value—a contradiction that fails if those growth bounds on H break.

What would settle it

Exhibit a single positive rational λ and arbitrarily large n for which some n-term Egyptian underapproximation (in either convention) strictly exceeds the value obtained by extending an optimal short terminal decomposition by its greedy unit-fraction tail.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Every positive rational eventually admits a unique greedy maximizing residual after a finite initial segment.
  • For rationals whose finite best underapproximations are already unique, any other infinite unit-fraction series summing to the same rational has strictly slower double-exponential denominator growth.
  • The same control formulation recovers the classical uniqueness theorems for unit fractions and for the special rationals treated by Nathanson and Chu as the base of the induction.
  • An explicit Liouville number is uniquely greedy at every finite length in both conventions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Bellman payoff may decide eventual greediness for some algebraic irrationals, the question left open by Erdős–Graham.
  • Quantitative bounds on n0(λ) would follow from effective estimates on the first descent of the cleared numerator and the size of Γ_P.
  • The completion-and-deletion procedure of Lemma 14 suggests a practical algorithm that certifies optimality of long greedy tails without enumerating all competing n-tuples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 7 minor

Summary. The paper proves that every positive rational λ has eventually greedy best Egyptian underapproximations in both the nondecreasing (σ=⩽) and strictly increasing (σ=<) denominator conventions (Theorem 1). The argument reformulates best underapproximation as an optimal-control problem on states (P,Q,L), constructs an explicit payoff H satisfying H(Φ(t))=2H(t) and log t ≤ H(t) ≤ log t + log 2, studies the associated Bellman functions B^σ, and proves finiteness and attainment by induction on the cleared numerator via first-descent blocks. A completion lemma then shows that any sufficiently good long competitor underapproximation yields a terminal decomposition whose discounted H-value cannot beat the optimal Bellman value, forcing eventual greediness. As applications, the authors obtain a growth comparison for unit-fraction series to certain rationals (Corollary 3) and construct a Liouville number whose best underapproximations are uniquely greedy at every length (Example 2).

Significance. The main theorem affirmatively settles a problem stated by Erdős–Graham and later posed as open by Nathanson, Graham, and others, for all positive rationals and under both standard denominator conventions. The optimal-control/Bellman reformulation is a genuine methodological contribution to Egyptian-fraction extremal problems; the payoff H is constructed explicitly (uniformly convergent series) and its key properties are proved elementarily, so the argument is self-contained rather than black-box. Corollary 3 converts known uniqueness theorems into sharp asymptotic comparisons (including the Vardi constant for λ=1), and Example 2 answers Nathanson’s question on irrationals with uniquely greedy best underapproximations. Strengths include full proofs of all lemmas, transparent AI-usage disclosure with public transcripts, and clean separation of the unit-fraction boundary case from the inductive step.

minor comments (7)
  1. [Title / p.1] Title page header reads “EVENTUALL Y GREEDY…” with a spurious space; fix throughout front matter.
  2. [§2.1] In §2.1 the informal sketch of the proof is helpful, but a one-sentence roadmap of which lemmas feed the final contradiction in §3.4 (H monotonicity → barrier 2^r H(T_k/2^r) ≥ H(T_k) → competitor contradiction) would make the long induction easier to navigate on first reading.
  3. [Lemma 8] Lemma 8 cites Karamata/Muirhead; a brief pointer that the needed majorization is only the weak form for the exponential would help readers less familiar with those inequalities.
  4. [Lemma 11] The constants Γ_k in Lemma 11 are defined recursively and later bounded by log(2k); stating the closed-form bound Γ_k ≤ log(2k) already in the definition would shorten several later displays.
  5. [§3.4, display (3.39)] In the strictly-increasing half of the proof of Theorem 1, the threshold conditions T_k > 8 and log T_k > 2n log 2 are chosen for convenience; a parenthetical remark that any fixed T_k > 4 and a slightly adjusted constant would also work would avoid the impression that 8 is canonical.
  6. [§4] Example 2 / §4 is clean; adding one sentence noting that the same Liouville construction works verbatim for both conventions (already proved) would make the example self-contained for readers who skip the nondecreasing induction.
  7. [References] References [16] and [21] are arXiv preprints from 2025; if journal versions exist by the time of revision, update the citations.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is proved from an independently constructed payoff H and a self-contained Bellman induction, not from the target claim or load-bearing self-citation.

full rationale

The derivation chain for Theorem 1 does not reduce to its inputs by construction. The payoff H is defined explicitly as the uniformly convergent series H(t)=log t + Σ 2^{-ℓ-1} log(1+1/t_ℓ) along Φ-iterates (Lemma 4) and is proved strictly increasing and log-bounded before any appeal to eventual greediness of rationals. Bellman values B^σ are defined as suprema over terminal decompositions; finiteness, attainment, and the stopped first-descent equation are obtained by strong induction on the cleared numerator (Lemmas 9–13), using only the classical unit-fraction boundary (Lemma 6, from Curtiss–Takenouchi/Nathanson/Chu) and elementary comparison (Lemma 8). Competing n-term underapproximations are completed to legal terminal decompositions (Lemma 14) and ruled out for large n by the independent barrier 2^r H(T_k/2^r) ≥ H(T_k) from (H1)–(H3) and doubly exponential growth of T_k. Author self-citations ([19], [20], [21]) supply motivation, measure-zero context, and the conditional pathway to Corollary 3; they are not premises inside the proof of Theorem 1. Example 2 is a direct inductive construction. No fitted parameters, no self-definitional loop, and no uniqueness theorem of the authors is used to force the main claim.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

Load-bearing background is standard arithmetic of unit fractions plus classical uniqueness/extremality for unit-fraction underapproximations and a few special rationals. The paper’s own constructions (payoff H, Bellman B^σ, first-descent blocks, discount 1/2 matched to Φ) are definitions with proved properties, not free fits. No empirical parameters.

axioms (6)
  • standard math Curtiss–Takenouchi uniqueness: greedy m-term underapproximations of 1 are uniquely best (both conventions).
    Invoked as Lemma 6 base for T=1 and unit-fraction boundary of B^σ.
  • standard math Nathanson uniqueness for reduced p/q with p | q+1 (includes 1/T, T≥2) in both conventions.
    Lemma 6 for T≥2; also feeds concrete cases of Corollary 3.
  • standard math Chu uniqueness under the odd-q / least-ℓ=2 condition for two-sided application remarks after Corollary 3.
    Used only to specialize Corollary 3; not required for Theorem 1 itself.
  • domain assumption Existence of maxima R^σ_n(λ) for finite n-term Egyptian underapproximations.
    Cited from Nathanson [23, Thm 3]; standard compactness/ discreteness fact for the problem setup.
  • standard math Product–sum comparison / weak majorization (Lemma 8) controlling cleared-numerator descent.
    Karamata/Muirhead-type inequality used to prove Lemma 9 bounds that drive the Bellman induction.
  • ad hoc to paper Choice of discount factor 1/2 per control matched to the functional equation H(Φ(t))=2H(t).
    Methodological design choice in §2–§3; proved consistent via Lemma 4, but the specific factor is part of the paper’s construction rather than an external theorem.
invented entities (2)
  • Payoff function H(t)=lim 2^{-j} log(Φ^{∘j}(t)) independent evidence
    purpose: Assign comparable terminal values so greedy unit-fraction tails preserve discounted payoff and competitors can be contradicted for large length.
    Constructed and proved to satisfy (H1)–(H3) in Lemma 4; not an external physical entity, but the central analytic device.
  • Bellman functions B^⩽ and B^< on states (P,Q,L) independent evidence
    purpose: Convert best underapproximation into dynamic programming over controls and terminal unit-fraction remainders.
    Defined as suprema of discounted H-values; finiteness/attainment proved by induction (Lemmas 11–13).

reviewed 2026-07-31 · how reviews work

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Cite this review

Pith. "Pith review of Eventually greedy best Egyptian underapproximations of rational numbers via optimal control." pith.science (2026). https://pith.science/paper/HX437XAU

@misc{pith2026260728387,
  author       = {Pith},
  title        = {Pith review of: Eventually greedy best Egyptian underapproximations of rational numbers via optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HX437XAU}},
  note         = {Machine review of arXiv:2607.28387}
}
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read the original abstract

We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erd\H{o}s and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.

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This paper was first reviewed by grok-4.5 on July 31, 2026.