Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Large Sets of Integers with No Harmonic Triples

T0 review · 2 major / 5 minor · reviewed 2026-07-08 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Large harmonic-triple-free sets match Behrend-type bound

desk verdict Clean, correct, and short. The paper proves the first Behrend-quality lower bound for harmonic-triple-free sets. The argument is sound and deserves a serious referee. read the letter →

arxiv 2607.05823 v1 pith:U6EMPYC2 submitted 2026-07-07 math.NT math.CO

Samuel Korsky This is my paper · ORCID
classification math.NTmath.CO MSC 11B2511B75
keywords harmonictriplesarithmeticprogressionsBehrendconstructionadditivecombinatoricsprogression-freesetsreciprocalmapfinitefields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

A harmonic triple is three distinct integers whose reciprocals form an arithmetic progression: 2/a = 1/b + 1/c. The paper proves that one can find a subset of {1,...,N} of size at least N exp(-(C+o(1)) sqrt(log log N)), with C = 2 sqrt(log(24/7)), containing no such triple. The construction takes a dense three-term-arithmetic-progression-free set in a prime field of size comparable to log N, uses a random affine image of it to filter the odd integers up to N, and then deletes elements that participate in surviving harmonic triples. The key observation is that any harmonic triple surviving the filter must collapse modulo the prime q, forcing its difference parameter to be divisible by q, which limits the number of such triples to O(N log N / q). A union bound over these collapsed triples shows that the deletions are absorbable, leaving a set of the claimed size. The bound has the same shape as the best known lower bound for progression-free sets, with log N replaced by log log N, reflecting the compression induced by the reciprocal map.

What carries the argument

Random affine images of progression-free sets in F_q, parametrization of odd harmonic triples via b=(h-d)hm, a=(h-d)(h+d)m, c=(h+d)hm, union bound over collapsed triples with q|d

What would settle it

If the count of collapsed triples U_q(Y) were substantially larger than Y log Y / q (for instance, if the parametrization missed a family of triples), the deletion step could remove too many elements and the construction would fail.

Watch

Extended reading notes

Core claim

The transfer mechanism works because the reciprocal map converts the additive structure of arithmetic progressions into the multiplicative structure of harmonic triples, but at the cost of compressing the effective scale from N to roughly log N. By working in a prime field of size q ~ log N, the author embeds a progression-free set into the residue classes of odd integers. The parametrization of odd harmonic triples (Lemma 2.2) shows that triples surviving the modular filter must have their difference parameter d divisible by q, making them rare enough to delete. The constant C_AP = 2 sqrt(log(24/7)) is inherited directly from the best known progression-free construction, and the log log N取代

Load-bearing premise

The construction relies on the external lower bound for progression-free sets r_3(X) >= X exp(-(C_AP+o(1)) sqrt(log X)) and on the union bound over collapsed triples being tight enough that the deletion term O(N log N / q) is absorbable when q is chosen large enough relative to log N.

Share X Bluesky LinkedIn Reddit HN

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

2 major / 5 minor

Summary. The paper proves that there exist subsets of [N] of size N exp(-(C_AP + o(1)) sqrt(log log N)) containing no harmonic triple (distinct a,b,c with 2/a = 1/b + 1/c), where C_AP = 2 sqrt(log(24/7)). The construction selects odd integers n <= N whose modular inverse n^{-1} mod q lies in a random affine image of a 3-AP-free set R in F_q (q ~ log N), then deletes the largest element of any surviving harmonic triple. The key observation is that a surviving triple must collapse to a 3-AP in R, forcing a = b = c mod q; a parametrization then shows q | d, limiting the number of such triples to O(Y log Y / q), which is absorbable.

Significance. The result cleanly transfers the best known 3-AP-free lower bound (Elsholtz–Hunter–Proske–Sauermann) to the harmonic-triple setting, with log N replaced by log log N reflecting the reciprocal map. The construction is explicit and the proof is short and self-contained apart from the cited external bound. The constant C_AP is inherited parameter-free from [4]. The open Question 1.2 (whether f(N) = o(N)) is a natural and well-motivated complement.

major comments (2)
  1. [Lemma 3.1, definition of B_0] The notation 'n^{-1} mod q' is written as 'n-1modq' in the definition of B_0 = {n in V : n-1 mod q in R_{lambda,mu}}. This is ambiguous and could be read as subtraction rather than modular inverse. The surrounding text (reduction of (1) mod q giving 2a^{-1} = b^{-1} + c^{-1}, the set V excluding multiples of q, and the phrase 'inverse residues') confirms the intended meaning is n^{-1} mod q. This should be stated unambiguously, as it is load-bearing for the entire argument: the 3-AP-free property of R_{lambda,mu} forces b^{-1} = a^{-1} = c^{-1}, hence a = b = c mod q, which is the key step.
  2. [Lemma 3.1, equation (12)] The inner sum bound in (12) writes sum_{1 <= j < h/q} 1/(h(h+qj)) << 1/(qh), using h+qj >= h. This is correct but the step should be slightly more explicit: since h+qj >= h for j >= 1, each summand is <= 1/h^2, and there are at most h/q terms, giving <= (h/q)(1/h^2) = 1/(qh). As written the inequality is valid but a reader might want this one-line justification, especially since the summability of 1/(qh) over h is what makes the deletion term O(delta Y log Y / q) = O(delta Y / K).
minor comments (5)
  1. [Lemma 2.1] The wrap-around argument states |x+z-2y| < q for x,y,z in [1, floor(q/3)]. Strictly, x+z-2y ranges in (-q/3, q), so |x+z-2y| < q holds. This is correct but could be stated more precisely.
  2. [Lemma 2.2] The parametrization requires b < a < c, but the converse direction notes that not all parameter choices yield odd solutions. The construction only uses odd integers (via V), so a brief remark that the parametrization still applies to the odd triples that arise in the deletion argument would improve clarity.
  3. [Introduction, equation (2)] The constant C_AP = 2 sqrt(log(24/7)) is described as 'the natural-logarithm form of the base-two constant appearing in [4]'. A brief note on the conversion (base-2 to natural log) would help readers verify the constant.
  4. [Lemma 3.1] The set V = {n <= Y : n odd, q does not divide n} has |V| >> Y, but the implicit constant depends on q being prime and odd. Since q >= K log Y, the fraction of multiples of q up to Y is 1/q = o(1), so |V| ~ Y/2. Making this explicit would be helpful.
  5. [References] Reference [4] is cited as a preprint (arXiv:2406.12290, 2024). If it has since been published or refereed, the citation should be updated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two notational/presentational issues in the proof of Lemma 3.1. Both points are well-taken and will be addressed in the revised manuscript. The mathematical content is unaffected; the revisions concern clarity of exposition only.

read point-by-point responses

  1. Referee: The notation 'n^{-1} mod q' is written as 'n-1 mod q' in the definition of B_0, which is ambiguous and could be read as subtraction rather than modular inverse.

    Authors: The referee is correct. In the definition of B_0 = {n in V : n^{-1} mod q in R_{lambda,mu}}, the superscript on n^{-1} was not properly typeset, making the modular inverse ambiguous with subtraction. This is indeed load-bearing for the argument: the 3-AP-free property of R_{lambda,mu} forces b^{-1} = a^{-1} = c^{-1} mod q, hence a = b = c mod q, which is the key step. We will rewrite the definition of B_0 to state unambiguously that n^{-1} mod q denotes the modular multiplicative inverse of n modulo q, and will add a brief sentence clarifying this convention when B_0 is first introduced. revision: yes

  2. Referee: The inner sum bound in (12) writes sum_{1<=j<h/q} 1/(h(h+qj)) << 1/(qh), using h+qj >= h. This is correct but the step should be slightly more explicit.

    Authors: The referee's suggested one-line justification is exactly right. Since h + qj >= h for j >= 1, each summand satisfies 1/(h(h+qj)) <= 1/h^2, and there are at most h/q terms in the sum, giving the bound (h/q)(1/h^2) = 1/(qh). We will insert this one-line justification into the display between the second-to-last and last lines of (12), so that the reader can directly verify the step leading to the O(Y log Y / q) bound. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found. The derivation is self-contained given an external benchmark.

full rationale

The paper's sole author, Korsky, does not cite any prior work by himself. The central constant C_AP = 2√(log(24/7)) is imported from Elsholtz–Hunter–Proske–Sauermann [4], a distinct group of authors, and is used as an external input (equation (2)). The main theorem (Theorem 1.1) is obtained by a genuine mathematical reduction: choosing q ≍ log N so that log q = log log N, embedding the external 3-AP-free set into F_q (Lemma 2.1), filtering odd integers by a random affine image, and showing that surviving harmonic triples either yield a 3-AP in the progression-free set (impossible) or collapse mod q with q|d (Lemma 2.2). The deletion count U_q(Y) ≪ Y log Y / q (equation (12)) is derived from first principles via the parametrization in Lemma 2.2, and the absorbability of the deletion term follows by choosing K large (equation (8)). No parameter is fitted to data and then presented as a prediction. No ansatz is smuggled in via self-citation. The derivation chain is clean and non-circular.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No new entities, particles, or structures are postulated. The construction uses standard mathematical objects (prime fields, affine images, arithmetic progressions).

free parameters (1)
  • K = sufficiently large absolute constant, chosen > 2c₂/c₁
    Controls the size of q relative to log Y. Not fitted to data; chosen large enough to ensure the deletion term is absorbed. The specific value does not affect the asymptotic bound.
assumptions (3)
  • standard math Bertrand's postulate: for all sufficiently large Y there is a prime q with K log Y ≤ q ≤ 2K log Y
    Used in Lemma 3.1 to guarantee existence of a prime of the right size. Standard result.
  • domain assumption Elsholtz–Hunter–Proske–Sauermann lower bound: r₃(X) ≥ X exp(−(C_AP+o(1))√(log X))
    External result [4], used in Lemma 2.1 to construct dense 3-AP-free sets in F_q. The present paper's contribution is the transfer, not this bound itself.
  • standard math Basic number theory: coprimality, divisibility, modular arithmetic used in Lemma 2.2
    Standard tools used in the parametrization of harmonic triples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Large Sets of Integers with No Harmonic Triples." pith.science (2026). https://pith.science/paper/U6EMPYC2

@misc{pith2026260705823,
  author       = {Pith},
  title        = {Pith review of: Large Sets of Integers with No Harmonic Triples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U6EMPYC2}},
  note         = {Machine review of arXiv:2607.05823}
}
abstract

Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\ldots,N\}$ containing no distinct $a,b,c$ such that \[ \frac2a=\frac1b+\frac1c . \] We prove \[ f(N)\gg N\exp\!\left(-(2\sqrt{\log(24/7)}+o(1))\sqrt{\log\log N}\right). \] The construction filters the odd integers up to $N$ by a random affine image of a dense three-term-progression-free set in a prime field $\mathbb{F}_q$ with $q\asymp\log N$, and then deletes a controlled family of collapsed triples.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sets of unit fractions without two members whose average is a unit fraction

    math.NT 2026-07 accept novelty 8.0 of 10

    For arbitrarily large N there is a set A of more than cN numbers a≤N such that for any distinct a,b in A, a+b does not divide 2ab—equivalently, the average of 1/a and 1/b is never a unit fraction.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

  1. [4]

    Improving Behrend's construction: Sets without arithmetic progressions in integers and over finite fields

    C. Elsholtz, Z. Hunter, L. Proske and L. Sauermann,Improving Behrend’s construction: Sets without arithmetic progressions in integers and over finite fields, arXiv:2406.12290, 2024

    work page Pith review arXiv 2024

  2. [1]

    F. A. Behrend,On sets of integers which contain no three terms in arithmetical progression, Proc. Natl. Acad. Sci. USA32(1946), 331–332, doi:10.1073/pnas.32.12.331

    work page doi:10.1073/pnas.32.12.331 1946

  3. [2]

    T. F. Bloom and O. Sisask,Breaking the logarithmic barrier in Roth’s theorem on arithmetic progressions, arXiv:2007.03528, 2020

    work page Pith review arXiv 2007

  4. [3]

    T. F. Bloom and O. Sisask,An improvement to the Kelley–Meka bounds on three-term arith- metic progressions, arXiv:2309.02353, 2023

    work page Pith review arXiv 2023

  5. [5]

    Elkin,An improved construction of progression-free sets, Israel J

    M. Elkin,An improved construction of progression-free sets, Israel J. Math.184(2011), 93–128, doi:10.1007/s11856-011-0061-1

    work page doi:10.1007/s11856-011-0061-1 2011

  6. [6]

    Erdős and P

    P. Erdős and P. Turán,On some sequences of integers, J. London Math. Soc.11(1936), 261–264, doi:10.1112/jlms/s1-11.4.261

    work page doi:10.1112/jlms/s1-11.4.261 1936

  7. [7]

    Green and J

    B. Green and J. Wolf,A note on Elkin’s improvement of Behrend’s construction, inAdditive Number Theory, Springer, New York, 2010, 141–144, doi:10.1007/978-0-387-68361-4_9

    work page doi:10.1007/978-0-387-68361-4_9 2010

  8. [8]

    Optimal fault-tolerant spanners in euclidean and doubling metrics: Breaking the Ω (log n) lightness barrier

    Z. Kelley and R. Meka,Strong bounds for 3-progressions, 2023 IEEE 64th An- nual Symposium on Foundations of Computer Science (FOCS), IEEE, 2023, 933–973, doi:10.1109/FOCS57990.2023.00059. 5

    work page doi:10.1109/focs57990.2023.00059 2023

Show all 10 references
  1. [9]

    K. F. Roth,On certain sets of integers, J. London Math. Soc.28(1953), 104–109, doi:10.1112/jlms/s1-28.1.104

    1953 doi

  2. [10]

    Salem and D

    R. Salem and D. C. Spencer,On sets of integers which contain no three terms in arithmetical progression, Proc. Natl. Acad. Sci. USA28(1942), 561–563, doi:10.1073/pnas.28.12.561. 6

    1942 doi

Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.