Conlon 2024 question erdos graham egyptian fractions

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claim_1: Proves the volume lower bound by integer divisor counting without assuming that the multiplier is a unit.

claim_2: Proves the strictly descending modular cancellation, with legal disjoint denominators and controlled reciprocal cost.

conditional_entropy: Reconstructs the source's conditional-entropy lower bound with uniform error estimates.

definitions: Fixes the entropy, multiplier, subset-sum, and powersmoothness conventions.

entropy_basics: Expands the chain rule, subadditivity, and support bound for finite distributions.

entropy_exponent: Proves existence, strict monotonicity, endpoint limits, and uniform scaling continuity of the exponent.

external_inputs: States the precise imported probability, subset-sum, and number-theory estimates.

finite_window: Gives a separate product-law proof of the counting lower bound in a window below the mean.

gap_symmetrization: Derives a proper symmetric progression and its volume bound from the exact positive-input CFP theorem.

lemma_1: Proves the entropy upper bound and the uniform lower bound after restriction to any denominator set.

lemma_2: Determines the entropy optimizer and proves the discrete-to-continuous estimates on the needed growing range.

lemma_3: Records the exact normal-approximation inequality imported by the entropy proof.

lemma_4: Records a counterexample to the printed general bound and proves the sufficient A at most q replacement.

lemma_5: Proves the density estimate by grouping each prime's exceptions under its first power above the cutoff.

moments: Proves the variance, third-moment, and coordinate-removal estimates without empty-slab assumptions.

reservoir_availability: Proves a uniform one-quarter density for legal auxiliary denominators and a sublinear raw reservoir bound.

reservoir_completion: Uses disjoint geometric Croot intervals to finish a positive remainder uniformly up to a logarithmic threshold.

source_versions: Separates the published 2025 proof, the retained 2024 manuscript, source corrections, and external inputs.

theorem_1: Combines the entropy upper bound and uniform absorption lower bound for each fixed positive rational.

theorem_2: Proves every residue has a short reciprocal-sum representation in the full epsilon range needed for absorption.

theorem_3: Records the imported positive-input proper-GAP theorem with its original witness scope.

theorem_4: Proves the uniform powersmooth-target counting lower bound with an explicit sufficient error tending to zero.

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David Conlon, Jacob Fox, Xiaoyu He, Dhruv Mubayi, Huy Tuan Pham, Andrew Suk, and Jacques Verstraëte, A Question of Erdős and Graham on Egyptian Fractions, Discrete Analysis 2025:28, 13 pp., journal page. Received 25 April 2024; published 19 December 2025. The article prints DOI 10.19086/da.154329, but Crossref registers that DOI to a different Discrete Analysis article.

The canonical published PDF is byte-identical to arXiv:2404.16016v2 (17 December 2025). The substantive earlier arXiv v1 (24 April 2024) is retained separately. See the source record and version and correction comparison. The existing 2024 folder name is retained as the canonical home. The arXiv record names the Creative Commons Attribution 4.0 license for conlon_2024_question_erdos_graham_egyptian_fractions.pdf (arXiv:2404.16016). The same record names that license for conlon_2024_question_erdos_graham_egyptian_fractions_arxiv_v1.pdf.

Theorem 1 proves that for each fixed positive rational $x$,

$$|\{A\subseteq[n]:\sum_{a\in A}1/a=x\}|=2^{c_xn+o_x(n)},$$

where $c_x\in(0,1)$ is the entropy integral defined and analyzed in entropy_exponent. It is continuous and strictly increasing from 0 to 1. This specifies the logarithmic growth rate, not a multiplicative asymptotic. The source reports $c_1\approx0.91117$; those digits are not numerically certified here. Since $c_1<1$, the theorem answers in the negative Erdős and Graham's question whether the count at $x=1$ is $2^{n-o(n)}$, recorded in Problem 297; the paper notes that the MathOverflow upper bound had already answered it. The published introduction attributes the earlier matching upper exponent to MathOverflow contributions; this compilation adds no historical priority or current-status claim.

The full local proof has two materially distinct stages. The finite entropy optimizer and its uniform limiting integral give the count below a reciprocal-sum threshold. The printed conditional-entropy route is reconstructed using complete moment estimates and finite entropy identities. A finite-window product-law alternative is separately labeled as a compilation deduction.

The exact-count lower bound then uses short modular reciprocal sums, proper symmetric progression reduction, inverse-pair counting, powersmooth supply, and descending prime-power cancellation, followed by Croot completion. The resulting uniform absorption theorem is proved with a sufficient error tending to zero. The fixed-rational limit takes $n\to\infty$ before that error parameter tends to zero.

The unrestricted printed Lemma 4 has an explicit counterexample; the sufficient restricted replacement and its prerequisite volume bound are fully proved. Other source precision corrections are recorded locally and in source_versions. They do not amount to a claim that the main theorem is false.

This unit reconstructs the complete main chain relative to exact external estimates from Berry–Esseen, CFP, Croot, the divisor bound, Dickman, and the prime number theorem. It does not compile those original proofs, the unused larger-parameter part of Theorem 2, numerical constants, or the distinct Liu–Sawhney counting proof. In particular, earlier coverage of Liu–Sawhney Theorem 1.1 does not supply their separate counting argument.

Bears on. Problem 297; Problem 148 (mentioned there to distinguish this denominator-cutoff count from the fixed-length count $F(k)$; no bound for $F(k)$ is drawn from it).