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The paper proves that every positive rational number has eventually greedy best Egyptian underapproximations in both denominator conventions, using Bellman-function methods from optimal control.

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2026-08-05 04:20 UTC pith:HX437XAU

load-bearing objection The paper proves the full Erdős–Graham–Nathanson conjecture for rationals in both denominator conventions via a genuinely new Bellman-function technique; the proof is long but appears sound, with only a well-supported classical dependency.

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 algorithmBellman functionoptimal controlrational underapproximationSylvester sequencedenominator growth
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.

The paper proves a conjecture originating with Erdős and Graham and later posed by Nathanson: for every positive rational number, once enough terms are taken, the best approximation from below by a sum of unit fractions is obtained by taking a best finite prefix and then appending greedy unit fractions one after another. This holds both when repeated denominators are allowed and when denominators must be distinct. The proof recasts the static extremal problem as an optimal control problem: the remainder after any partial sum is a state, each new denominator is a control, and a carefully built payoff function H with H(t(t+1)) = 2H(t) makes greedy steps value-preserving. A Bellman function bounds the value of every possible completion and, by induction on a cleared numerator, is shown to be finite and attained; the maximizing completions are eventually unique and coincide with the greedy tail. As applications, the paper identifies the maximal double-exponential growth rate of denominators in any unit-fraction series converging to a rational, and constructs an irrational number whose best underapproximations are unique and greedy at every length.

Core claim

The central result, Theorem 1, states that for σ = 'nondecreasing' or 'strictly increasing' denominators, every positive rational λ has a threshold n0 such that for every m ≥ 1 the best (n0+m)-term underapproximation equals the best n0-term underapproximation plus the best m-term underapproximation of the remainder, and the greedy m-term tuple is the unique maximizer for that remainder. The proof recasts the problem as optimal control: a state (P,Q,L) records the current remainder and the largest denominator already used; each new denominator x is a control leading to (Px−Q, Qx, x). Terminal completions end in a unit fraction 1/T and are valued at 2^{−m}H(T), where H is the unique increasing

What carries the argument

The Bellman function B^σ(P,Q,L) = sup 2^{−m}H(T) over terminal decompositions, with payoff H the unique increasing solution of H(t(t+1)) = 2H(t) satisfying the logarithmic bounds log t ≤ H(t) ≤ log t + log 2. The self-similarity of H makes appending a greedy unit fraction value-preserving, so optimal paths eventually follow the forced Sylvester-type tail; induction on the 'cleared numerator' shows the Bellman value is finite and attained, and the classical unit-fraction extremality theorem supplies the obstacle boundary that anchors the induction.

Load-bearing premise

The proof rests on the known theorem that for 1/T the greedy m-term sums are the unique best m-term underapproximations; if that failed, the boundary condition anchoring the Bellman induction would collapse.

What would settle it

Choose a rational λ, e.g. 10/61, fix a denominator convention, and compute the best n-term underapproximation for all n up to a large bound by exhaustive search over denominators below a growing cutoff; the theorem predicts that for n ≥ some n0, the best (n+1)-term tuple is obtained by appending the greedy denominator to a best n-term tuple. Any rational for which these disagree at arbitrarily large n disproves Theorem 1; a more local test is to find an admissible state (P,Q,L) with P ≤ Q whose Bellman supremum is not attained.

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If this is right

  • For every rational λ, the greedy algorithm is eventually optimal in both denominator conventions, so any initial deviations from optimality disappear.
  • The maximal double-exponential growth rate of denominators in a unit-fraction series summing to a rational is identified: every non-optimal expansion has a strictly smaller liminf of a_n^{2^{−n}}, unless it is eventually identical to the greedy one.
  • There exist irrational numbers, of Liouville type, whose best n-term underapproximations are unique and greedy for all n.
  • The Bellman-function proof supplies a uniform structural explanation for previously isolated cases where greedy and best approximations coincide, and it recovers the classical Sylvester-type growth constant for λ = 1.

Where Pith is reading between the lines

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

  • The method could plausibly be adapted to algebraic λ if a substitute for the unit-fraction boundary condition can be found; the paper explicitly leaves the algebraic variant open.
  • Because the Bellman function is attained and has finitely many maximizing rays, the threshold n0 and the optimal prefix are in principle computable by dynamic programming over cleared numerators.
  • The dichotomy of Theorem 4 suggests a quantitative stability statement: any expansion that tracks the optimal double-exponential rate for a long prefix is forced to equal it, which could be tested on small-denominator rationals.
  • The payoff function H, given by an infinite series over iterates of t ↦ t(t+1), may itself be a useful growth gauge for Sylvester-type dynamics beyond Egyptian fractions.

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 / 3 minor

Summary. This paper proves Theorem 1: for every positive rational number λ, and for both the nondecreasing (σ=≤) and strictly increasing (σ=<) denominator conventions, the best n-term Egyptian underapproximation is eventually greedy. Explicitly, there exists n0 = n0(λ, σ) such that R^σ_{n0+m}(λ) = R^σ_{n0}(λ) + R^σ_m(λ − R^σ_{n0}(λ)) for all m ≥ 1, with the greedy m-term denominator tuple uniquely maximizing the remainder. The proof recasts the problem as an optimal control problem: states are triples (P, Q, L) representing the current remainder and the last denominator; terminal decompositions have payoff 2^{-m}H(T), where H is a carefully constructed strictly increasing function satisfying H(t(t+1)) = 2H(t) and log t ≤ H(t) ≤ log t + log 2. The associated Bellman functions B^σ are shown finite and attained via strong induction on the cleared numerator, and a completion lemma (Lemma 16) compares arbitrary competing underapproximations with greedy extensions. The same framework yields Theorem 4 on the maximal double-exponential growth of denominators in unit-fraction expansions of rational λ, and Example 2 constructs a Liouville number with unique greedy best underapproximations. The paper also proves Corollary 3 and other applications of the main result.

Significance. If the proof is correct, this resolves a conjecture of Erdős–Graham–Nathanson in both denominator conventions and supplies the missing ingredient for the asymptotic extremality results in Corollary 3 and Theorem 4. The optimal-control/Bellman-function approach appears to be new for Egyptian-fraction extremal problems and is executed in detail: H is constructed explicitly with its monotonicity and exact-doubling properties proven; the induction on cleared numerators is the natural descent argument; and the completion lemma cleanly converts Bellman optimality into fixed-length optimality. The reliance on the classical unit-fraction extremality theorem (Lemma 8) is genuine but well documented and is not an internal gap. No machine-checked formalization is provided, but the analytic proof is essentially self-contained apart from standard cited results. I found the proof long but coherent; the main caveats are presentation issues in Section 4 and a slightly compressed treatment of some applications.

minor comments (3)
  1. [Section 4 (Theorem 4 proof)] The sentence 'Since d_j→∞, one has c_{k+j}=d_j for all sufficiently large j' is not literally correct when finitely many d_j are smaller than the fixed denominators a_1,...,a_k. After merging, one only gets c_{s+j}=d_{r+j} for large j, with s−r=k. The displayed identity lim_n (c^{(k)}_n)^{2^{-n}} = D_k^{2^{-k}} remains true because the exponent shift is s−r=k, but the text should be reworded to give the index-offset argument precisely. As written, a careful reader may think the proof relies on a false equality of indices.
  2. [Section 3, Lemma 8 and boundary condition (3.8)] The uniqueness statement in Lemma 8 is load-bearing, and for T≥2 the proof cites Nathanson [23, Theorem 5] and Chu [8]. Since [23] is primarily formulated for nondecreasing denominators, it would be helpful to add a sentence explaining explicitly that the same greedy tuple is the unique maximizer in the strictly increasing convention as well, or to point to the precise statement in [8] that covers this case. This is a citation-clarity issue, not a correctness concern.
  3. [Section 5, Example 2] The proof of Example 2 is clear, but the notation b_n = 2^{n!} and the later use of b_n for the denominators of the greedy tail in Corollary 3 and Theorem 4 could be confused. A brief remark that the b_n in Example 2 are unrelated to the b_n in Corollary 3 would prevent a possible misunderstanding.

Circularity Check

0 steps flagged

No circular derivation: central proof is an independent Bellman-function argument; the only external load-bearing input is classical unit-fraction extremality, and self-citations are contextual.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. The payoff function H is constructed in Lemma 5 directly from the map Phi(t)=t(t+1), with existence, uniqueness, monotonicity, and logarithmic bounds proved in the text; it is not defined in terms of best underapproximations for general rationals. The Bellman functions B^sigma are defined as suprema over terminal decompositions with value 2^{-m}H(T), and the proof that they are finite and attained is carried out by strong induction on the cleared numerator (Lemmas 11-15), using only arithmetic estimates and the previously proved properties of H. The comparison with arbitrary n-term underapproximations is made through Lemma 16, whose completion and collision-deletion argument is proved in the paper, not assumed. The only external load-bearing input is Lemma 8, the classical unit-fraction extremality theorem of Curtiss-Takenouchi, Nathanson, and Chu, which supplies the boundary condition B(P,Q,L)=H(Q/P) when Q/P is an integer. That theorem is independent, well established, and not a self-citation; it is also not equivalent to the theorem being proved, since the main difficulty is passing from unit-fraction remainders to arbitrary rationals. The self-citations ([19], [20], [21]) appear as context: a measure-zero result, previously posed conjectures that are now resolved by Theorem 1, and an adapted proof that is in fact re-proved in the text. None of these is load-bearing for the central claim. The AI-usage declaration reports provenance and drafting assistance but does not assert that any theorem is assumed or that any step in the derivation chain is missing, so it does not affect the circularity verdict. No equation or construction reduces by definition to the conjecture being proved; in particular, no fitted parameter is renamed as a prediction and no uniqueness theorem is imported from the authors' prior work to force the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters or invented entities are introduced. The central claim is derived from the constructed payoff function H (proven in Lemma 5) plus classical background facts. The only external dependencies are standard theorems in Egyptian fractions and convex analysis, listed above.

axioms (5)
  • standard math For every integer T, the greedy m-term tuple uniquely maximizes R_σ_m(1/T) (Curtiss-Takenouchi, Nathanson, Chu).
    Used in Lemma 8 and as the Bellman boundary condition (3.8); it is cited from prior literature, not proved in the paper.
  • standard math Best n-term Egyptian underapproximations exist: the maximum defining R_σ_n(λ) is attained.
    Invoked in the introduction and in Lemmas 7 and 15; cited to Nathanson's Theorem 3.
  • standard math The greedy algorithm applied to a rational remainder terminates in finitely many steps, reaching a unit fraction.
    Used throughout Sections 2 and 3 (e.g., Lemmas 12, 15, 16); this is the classical Fibonacci greedy algorithm for rationals.
  • standard math Karamata's inequality (or Muirhead's inequality) applies to the weak majorization step in Lemma 10.
    Used to prove the product-sum comparison behind the first-descent estimates; the paper cites [17] for Karamata's inequality.
  • standard math The harmonic series diverges, so for any λ > 1 a finite initial block of denominators can be chosen with remainder below 1.
    Used in Lemma 15 to show existence of a terminal decomposition for initial remainders P/Q > 1.

pith-pipeline@v1.3.0-alltime-deepseek · 23874 in / 12864 out tokens · 134611 ms · 2026-08-05T04:20:51.506984+00:00 · methodology

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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.

Figures

Figures reproduced from arXiv: 2607.28387 by Quanyu Tang, Vjekoslav Kova\v{c}.

Figure 1
Figure 1. Figure 1: Graph of H. denominator collides with an original denominator, delete the colliding original term and restart. After r ⩽ n deletions, this produces a modified error h ′ satisfying h ′ ⩽ 2 rh and a terminal decomposition for which B(p, q, 0) ⩾ 2 −(n−r)H(1/h′ ). Suppose now that n = N + k and that an n-term underapproximation is better than the above greedy extension. Its error satisfies h < 1/Tk, and the co… view at source ↗

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