Liu 2024 further questions regarding unit fractions

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counting_carrier: Finds a coprime integer with few prime factors and lifts it to an admissible multiple of the prescribed divisor.

counting_period_obstruction: Gives a two-adic counterexample to the printed unrestricted target period while preserving the counting theorem's target one.

evidence/: Retains the reviews of 2026-09-05 (main proof, ruled a coordinated check on 2026-09-18; preliminary results; bounded source checks; sampled supplement) and the independent fresh main-proof review and grade of 2026-09-18.

fact_2_5: Gives the Taylor expansion and absolute-value bound used for major and minor Fourier arcs, with the source's quadratic typo identified.

lemma_2_2: Records the false printed counting bound and a sufficient reciprocal-mass deduction used in the quantitative unit-subsum proof.

lemma_2_3: Bounds the number of integers up to N with a prime-power divisor larger than N divided by t.

lemma_2_4: Gives a uniform estimate for the integers in an interval avoiding a prescribed set of sufficiently small prime divisors.

lemma_2_6: States the bounded-increment martingale concentration inequality used to control random reciprocal sums.

lemma_3_1: Bounds the major-arc contribution below by three quarters of the reciprocal common denominator for a sufficiently large denominator set.

lemma_3_3: Proves the density bound using the valid reciprocal-mass estimate and the prime-power deletion lemma.

lemma_5_1: Extracts multiples of a large divisor with controlled reciprocal mass; distinguishes the false unrestricted statement from the proved form used.

lemma_6_1: Localizes reciprocal mass to an interval with logarithmic relative width.

lemma_6_2: Retains almost all reciprocal mass while making every surviving prime-power fiber heavy.

proposition_1_4: Dense sets yield a reciprocal subsum with denominator at most exp(C/alpha).

proposition_3_2: Proves the exact-event probability needed for counting with the actual denominator period and a stronger logarithmic parameter condition.

proposition_5_2: Gives a Fourier and sieve criterion for representing x/Q by a subset; gives the full proof with explicit corrections to the v1 parameter checks.

theorem_1_1: A reciprocal mass of at least (log N)^(4/5+epsilon) forces a subset summing to one.

theorem_1_2: Determines the exponential growth rate of subsets of [1,N] whose reciprocal sum is one.

theorem_1_3: A set of at least (1-1/e+epsilon)N denominators contains a unit subsum.

theorem_1_5: Represents a/b with largest denominator of order b log b times iterated logarithms.

theorem_1_6: Bounds the least denominator that cannot begin a unit representation ending by N.

theorem_2_1: Records the standard prime-number estimates used in Liu and Sawhney's smoothness, sieving, and common-denominator arguments.

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Yang P. Liu and Mehtaab Sawhney, On further questions regarding unit fractions. arXiv:2404.07113v1 (10 April 2024), 22 pages. Published in International Mathematics Research Notices 2026(2), rnaf382 (14 January 2026), DOI 10.1093/imrn/rnaf382.

Source and version

The copy read for this card is the 2024 arXiv version, arXiv:2404.07113v1. The arXiv record listed only v1 when checked on 2026-09-05 and again. The publisher record confirms the same abstract bound and a later publication, but its full text has not been compared. That record dates the article received 28 October 2025, accepted 23 December 2025 and published online 14 January 2026. All labels and page numbers below refer to v1; they must not be silently transferred to the published version. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2404.07113), every other right reserved.

The digest was checked against the PDF and corrected during result compilation. In particular the reciprocal-mass extremal function is not the minimum-term quantity $k(N)$ in Problem 295, and Theorem 1.6 matches Problem 294 despite the paper's nearby reference to 305.

Results and method

Theorem 1.1 states that, for every fixed $\varepsilon>0$ and sufficiently large $N$, a set $A\subseteq[1,N]$ with reciprocal mass at least $(\log N)^{4/5+\varepsilon}$ contains a subset summing to one. It improves Bloom's quantitative bound. The introduction (p. 1) attributes the earlier bound of order $(\log N)(\log\log\log N)/\log\log N$ for the extremal reciprocal mass of a set avoiding a unit subsum, and a lower bound of order $(\log\log N)^2$, to prior work; these are cited background. For fixed $\delta>0$ its threshold is below $\delta\log N$ for large $N$, so it also answers Problem 47, which Bloom's Theorem 3 had already settled. The paper refines the Fourier methods of Croot and Bloom. Its main technical proposition separates reciprocal mass from divisor incidence, and a sieve controls exceptional primes (pp. 4–6, 16–19).

The other main results have different compilation scopes. The counting theorem has a complete rewritten proof through the restricted Proposition 3.2. The remaining result pages below are statements and proof pointers whose full proofs are not yet compiled; the problem each bears on is named in parentheses:

• Theorem 1.2: the count of subsets with reciprocal sum one is $\exp((\gamma_*+o(1))N)$, with $\gamma_*\approx0.631573$ defined by an integral equation (Problem 297).
• Theorem 1.3: the sharp asymptotic cardinality threshold is $(1-1/e)N$ (Problem 300).
• Proposition 1.4: density $(\log N)^{-1/7+\varepsilon}\le\alpha\le1/2$ yields a positive reciprocal subsum at most one whose denominator is at most $\exp(C/\alpha)$ (Problem 310).
• Theorem 1.5: every proper fraction $a/b$ has a representation whose largest denominator is at most $b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$ (Problem 305).
• Theorem 1.6: the least forbidden first denominator is between $N/((\log N)(\log\log N)^3(\log\log\log N)^{O(1)})$ and $O(N/\log N)$ (Problem 294).

The simpler Fourier argument of Proposition 3.2 is used for the counting and denominator results, where the ambient set can be chosen; the main Proposition 5.2 is designed for arbitrary input sets. For counting, the actual-period, sixth-power form of Proposition 3.2 and Lemma 3.3 now have complete rewritten proofs. The full proof of Lemma 4.1 and the later denominator applications remain to be compiled at their exact target and parameter scopes. Earlier alternative methods and later refinements must be checked there before calling any of their bounds current best.

Proof coverage and source concerns

Theorem 1.1 has a complete outer reduction with parameter checks. Its essential preliminary and technical results have separate pages, with external prime-number, sieve and concentration inputs identified. Local typesetting corrections are explicit, including Lemma 6.1's missing minus sign, Lemma 6.2's unbound lowercase $n$, and the Fourier Taylor coefficient in Fact 2.5 and Lemma 3.1.

The complete main proof now uses explicitly justified forms of its inputs. The literal multiplicity-counting statement in Lemma 2.2 is false; that page records a counterargument and a separately proved Euler-product deduction giving the sufficient reciprocal loss $O((\log N)^{-\beta})=o(1)$, with $\beta=5\log(3/2)-3/2>0$. The literal unrestricted statement of Lemma 5.1 also fails for very small mass parameters. Its page preserves that limitation and proves the $H\ge2$ application form with corrected divisor exponents and the global prime-factor bound.

[[unit_fractions/liu_2024_further_questions_regarding_unit_fractions/proposition_5_2|Proposition 5.2]] proves that its feasible parameters satisfy $H\to\infty$ and lie in the divisor lemma's required range. It also tracks the dyadic endpoints and logarithmic powers, retains the missing Fourier factor, and supplies the omitted major-arc cardinality argument. Independent reviews checked these bounded corrections before incorporation, retained as the source checks and preliminary review. The exact revised main-proof pages, as they stood at 2026-09-18T07:24:04Z, passed independent blind review on 2026-09-18, retained as the fresh main-proof review with its distinct grade, which records PASS for the report contract and PASS for independence; the earlier main-proof review of 2026-09-05 is retained as a coordinated compilation check whose main-proof leg was ruled not independent on 2026-09-18, because its reviewer read the sibling reports' verdicts on steps inside its subject. These are explicit compilation corrections, not author errata and not assertions about the uninspected published version. They preserve Theorem 1.1's statement and do not affect Bloom's independent resolution of Problems 298 and 299.

The counting chain is separate from that arbitrary-input proof. The literal period in v1 Proposition 3.2 has a two-adic counterexample; its compiled replacement uses $Q=\operatorname{lcm}(A)$ and the sufficient condition $S\le K^3/(CN^2(\log\log N)^6)$. The proof supplies the required averaging range, set-valued complementary residue fibers, bounded prime choices, a full integer-carrier construction, and the cyclic frequency count. Theorem 1.2 uses the triple-log cutoff already printed on source p. 12, normalizes by the reciprocal of the actual mean, and tracks the uniform atom probabilities. Its global $N$ corrects the lowercase $n$ in the p. 13 logistic-odds display. The conclusion is the exact exponential rate for both sum one and sum at most one; the displayed decimals remain source approximations. The literal general proposition and its full fifth-power range are not claimed. PNT, Bertrand's postulate and Azuma–Hoeffding remain explicit external inputs.

A targeted literature search found no later improvement of Theorem 1.1's exact-sum reciprocal-mass threshold, and a further check of the citing-paper records for this paper and of the arXiv listings under unit and Egyptian fractions found no later improvement of Theorems 1.1, 1.3 or 1.6. The 2026 preprint arXiv:2607.04157 concerns approximate subsum one for multisets, a different question, treated on its own library card. No existing formalization of the stronger Liu–Sawhney threshold was located; the existing Bloom formalization proves the earlier theorem.

Relation to E301

This source bears on Problem 301.

For Problem 301, fix $a\in A$ and put $L=\lfloor N/a\rfloor$ and $C_a=\{m\in\{2,\ldots,L\}:am\in A\}$. The forbidden equation $1/a=\sum_i1/b_i$ with distinct elements of $A$ holds whenever some subset of $C_a$ has reciprocal sum $1$: if $\sum_{m\in B}1/m=1$ with $B\subseteq C_a$, then $1/a=\sum_{m\in B}1/(am)$ with the distinct elements $am\in A$, each exceeding $a$. Thus Theorem 1.1 (p. 1) implies that an E301-free $A$ satisfies $\sum_{m\in C_a}1/m<(\log L)^{4/5+\varepsilon}$ for every fixed $\varepsilon>0$ once $L$ is sufficiently large. Theorem 1.3 (p. 2) similarly gives $|C_a|<(1-1/e+\varepsilon)L$. These are restrictions on the multiples of each $a\in A$; neither is a global upper bound near $N/2$ for the extremal function $f(N)$ of Problem 301. The converse fails, since a solution need not consist of multiples of $a$ ($1/2=1/3+1/6$).

Proposition 5.2 (p. 16) could enter a direct forcing argument by applying it to suitable denominators in $A\setminus\{a\}$ with target $x/Q=1/a$. This requires, among its other hypotheses, $a\mid Q$ and the prescribed reciprocal sum range, so an arbitrary dense $A$ is not covered. Theorem 1.2 and Theorem 1.6 (pp. 2–3) concern representations of $1$ and do not themselves force an E301 identity. The source is relevant because it provides strong representation tools for the scaled sets $C_a$, but it does not prove $f(N)=(1/2+o(1))N$ or resolve Problem 301.

Bears on

• Problem 47
• Problem 294
• Problem 297
• Problem 298
• Problem 299
• Problem 300
• Problem 301
• Problem 305
• Problem 310

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.