Source versions and proof qualifications

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 7

The canonical 13-page PDF is the published Discrete Analysis 2025:28 article. Its first page prints DOI 10.19086/da.154329, which Crossref registers to a different Discrete Analysis article. Its first page says received 25 April 2024 and published 19 December 2025. The freshly acquired arXiv:2404.16016v2 PDF, dated 17 December 2025, is byte-identical to that retained canonical file.

The 8-page arXiv v1, dated 24 April 2024, is preserved separately. All 13 canonical pages and all 8 v1 pages were visually read. The following is a version comparison, not a claim of proof equivalence.

Subject	Published/v2 label and pages	v1 label and pages
Exact exponential count	Theorem 1, 2, 11	Theorem 1, 1, 8
Entropy counting bound	Lemma 1, 2, 4–7	Lemma 2, 2–5
Optimizer and multiplier	Lemma 2, 3–4	Lemma 3, 2–3
Berry–Esseen input	Lemma 3, 4	Lemma 4, 3
Modular reciprocal sums	Theorem 2, 7–8	Theorem 5, 5–6
CFP input	Theorem 3, 7	Theorem 6, 5
Modular approximation	Lemma 4, 7–8	Lemma 7, 5
Inverse-pair count	Claim 1, 8	Unnumbered claim, 6
Powersmooth supply	Lemma 5, 9	Lemma 9, 7
Uniform absorption	Theorem 4, 9–11	Theorem 8, 6–8
Prime-power cancellation	Claim 2, 10	Unnumbered claim, 7

The versions have substantive differences. The v1 introduction credits Steinerberger's upper exponent 0.93; the published introduction credits 2017 MathOverflow contributions by Lucia, RaphaelB4, and js21 for the matching upper exponential constant. These are the source's historical attributions; their separate original arguments are not compiled here.

In v1 p. 5 the homogeneity divisibility is reversed; the published p. 7 uses the correct condition $\gcd(d_1,\ldots,d_k)\mid x_0$ . The v1 third-moment display on p. 3 lacks absolute values; published p. 5 uses absolute third moments. The published Theorem 4 explicitly includes $\varepsilon\le x$ , whereas v1 Theorem 8 omits that positive lower bound. The remainder invariant on v1 p. 7 and the terminal $K(x-x_f)$ quantity on v1 p. 8 are corrected in the published text to the remainder $x-s(A_i)$ and the terminal $Kx_f$ , respectively. The published subset-sum interval on p. 8 has upper endpoint $sq/2$ ; the earlier display on v1 p. 6 uses $sq$ . Only the canonical version supplies the theorem labels used in this unit.

The compilation makes the following repairs or expansions to the published proof, each detailed at the linked result.

lemma_2 uses strict concavity, handles the finite optimizer's endpoint regimes, and restricts the multiplier estimates to a fixed positive lower threshold. Its Riemann comparison is uniform on the growing range.
moments replaces potentially empty integer slabs by integral and variation estimates, including removal of any coordinate.
conditional_entropy retains the necessary $1+t$ entropy term and controls the cutoff and conditional distribution uniformly. finite_window is a separately labeled compilation alternative; it is not credited as the source's printed method.
lemma_1 sends counted sets to their intersections with $U$ , correcting the printed final complement.
lemma_4 gives a counterexample to the unrestricted printed formula and a complete sufficient replacement for $A\le q$ . gap_symmetrization proves this volume prerequisite in dimensions at least 2, preserves properness under the smaller symmetric dilation, and first converts the centered input to a positive CFP input.
theorem_3 agrees with the original CFP theorem that the small witness is contained in the large retained set. They have different roles but there is no source error in that inclusion.
claim_1 retains integer endpoint terms and does not presume multiplication by the modular multiplier is injective. theorem_2 separately handles the unit step in dimension 1.
reservoir_availability proves the sufficient density $1/4$ in place of the printed $1/2$ , and claim_2 uses the positive cancellation congruence and proves disjointness of the selected, rather than raw, reservoir pieces.
lemma_5 expands the valid minimal-prime-power grouping and handles the factor $1/2$ in the cutoff with parameter $2\varepsilon$ . reservoir_completion gives integer disjoint Croot intervals. theorem_4 fixes the uniform choice order and obtains the sufficient error $8\varepsilon\to0$ .

These are compilation deductions and source-reading corrections, not a claimed author erratum or a claim that the main theorem is false. The complete local chain proves the main theorem relative to the exact external inputs. Berry–Esseen, CFP, Croot, the divisor estimate, Dickman's asymptotic, and the prime number theorem are not proved in this unit. The unused $\varepsilon\ge1$ range of printed Theorem 2 and the false unrestricted Lemma 4 are not certified.

The numerical integral evaluations, possible lower-order or multiplicative asymptotics, and the distinct Liu–Sawhney counting argument remain outside the full-proof scope. No current-status search, public formal acceptance check, or local formal build is claimed by this source compilation. Bibliographic acquisition and version identities are recorded in source_record.json.

Bears on. Problem 297.

Linked from (4)

Problem 297 Exact exponential rate of the unit-sum count unit_fractions/conlon_2024_question_erdos_graham_egyptian_fractions Counting distinct unit fractions

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