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

Will Sawin

REVIEW 2 major objections 4 minor 10 references

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

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read For every large N, a subset of {1,…,N} of size > cN exists in which no two unit fractions have a unit-fraction average, negatively answering a question of Erdős and Graham.

desk verdict A clever positive-density construction that settles an Erdős–Graham question, pending verification of one external dyadic estimate — worth refereeing. read the letter →

arxiv 2607.15419 v1 pith:EPFL7DBL submitted 2026-07-16 math.NT math.CO

Will Sawin This is my paper · ORCID

classification math.NTmath.CO MSC 11B0511N2511N37

keywords unitfractionsErdős–Grahamproblemaveragesofprimefactorscountedwithmultiplicitymultiplicativefunctionsmeanvaluetheoremsthree-termarithmeticprogressionspositivedensity

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

The paper proves that a question of Erdős and Graham, which asked whether every set of integers up to N with no two whose reciprocal average is a reciprocal must have vanishing density, has a negative answer. The construction is explicit: take all a ≤ N for which no b with Ω(b) ≤ Ω(a) and a ≠ b satisfies a+b | 2ab. The argument shows this set contains at least cN elements for some c > 0, so it has positive lower density. A direct consequence is the best known lower bound for the size of a set of unit fractions with no non-trivial three-term arithmetic progression.

What carries the argument

The proof uses the change of variables u = a/gcd(a,b), v = b/gcd(a,b), which turns a+b | 2ab into u(u+v) | 2a together with the conditions v ≤ uN/a, Ω(v) ≤ Ω(u), and gcd(u,v)=1. The density of a in S divisible by u(u+v) is bounded by a pointwise estimate involving multiplicative functions g1(u)g2(v)g3(u+v), and the total is controlled by a dyadic sum bound obtained from a cited mean-value theorem for nonnegative multiplicative functions over the binary quadratic form X1 X2 (X1+X2). The key inequality is that the exponent (1+ε) log 4 − 5/2 is less than −1 because log 4 < 3/2, which makes the final dyadic series converge.

What would settle it

Verify numerically for one dyadic interval, say u ∈ [X,2X] with X = 10^6 and L chosen so X ≥ L/(2(1+δ^{-1})), whether Σ_{g1(u)g2(v)g3(u+v)} ≤ C X² (log X)^{-7/4} (log L)^{-3/4}; if the sum exceeds this bound by a constant factor times a positive power of log X, Lemma 6 is refuted, and Theorem 1 collapses.

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Extended reading notes

Core claim

The central claim is that the set A_N defined by the divisibility condition a+b ∤ 2ab for all b with Ω(b) ≤ Ω(a) has positive lower density. The proof works by restricting to a carefully chosen set S of numbers with no small prime factors and no excess of medium prime factors, showing the average number of forbidden pairs per element of S is less than 1/2, and concluding |A_N| > |S|/2 > cN. This negative answer disproves the density-zero conjecture implicit in the Erdős–Graham question.

Load-bearing premise

Lemma 6, the load-bearing step, is obtained entirely from a cited external mean-value theorem applied with k=3, t=2 and polynomials X1, X2, X1+X2; if that theorem does not cover this case, or the estimated Euler product for that sum is wrong at primes 2 and 3, Lemma 6 fails and with it the positive-density conclusion.

Editorial extensions

If this is right

The Erdős–Graham density-zero question is answered negatively: there is a positive-density set A_N for which a+b ∤ 2ab for all distinct a,b.
The reciprocals of A_N form a set of unit fractions with no non-trivial three-term arithmetic progression, yielding the best known lower bound for that problem.
The construction is effective: for all sufficiently large N, |A_N| > cN with an explicit rule, and the proof gives a lower bound for c (not computed).
The same method gives a lower bound for the variant a+b ∤ ab, though weaker than the trivial odds construction; an optimized version might resolve that question.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same framework could be adapted to the stronger condition a+b ∤ ab (the sibling Erdős–Graham problem), and an optimized version might beat the trivial odd-number construction; this is a testable extension.
The construction's explicit nature means the constant c could be made effective with sufficient bookkeeping; a likely follow-up is determining how large c can be, possibly guided by numerical experiments on small N.
The exponential base 4 in the multiplicative functions is tied to the convergence condition log 4 < 3/2; changing the base changes the exponent of the final log, suggesting the barrier for this method is the constant (1+ε) log 4 − 5/2 < −1, beyond which new ideas are needed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: the paper answers Erdős–Graham's 1980 question by giving an explicit set A_N of positive lower density with no two elements satisfying a+b | 2ab. The construction is genuinely new: keep a only when no b with Ω(b)≤Ω(a) witnesses the divisibility. That Ω comparison is what keeps the pair count bounded, and it is the entire trick. The proof is otherwise a clean reduction to a dyadic sum of multiplicative functions; Lemmas 2–5 are correct, Lemma 4 is a standard Rankin-type estimate, and the final convergence only needs log 4 < 3/2.

The soft spot is exactly the one the reader and the stress-test note flag. Lemma 6 is a direct invocation of de la Bretèche–Tenenbaum [3, Theorem 3.1] for Q=X1X2(X1+X2), t=2. The manuscript lists the choices and then asserts that the hypotheses are satisfied without checking them. The p=2 issue is real: ρ_Q^+(2)=4=2^2, which could violate a strict inequality condition if the theorem has one. The author's local calculation at p=2 is meant to absorb this into E_R, and the L-dependence of F is also handled term by term, but the paper should say explicitly that the theorem allows finitely many exceptional primes or else give a reference to the relevant hypothesis. If the theorem does not allow this, the exponent -7/4 in Lemma 6 fails and with it the density claim. That is the one load-bearing external step, so a referee needs to open [3] and verify.

I want to be clear that I don't see this as a flaw in the main idea. The construction is elegant, the proof is honest, and there is no circularity or fitted constant anywhere. The narrative about Cambie, Stef, and the AI use is transparent and doesn't affect the mathematics. The paper deserves a serious referee. My recommendation: send it to an analytic number theorist, and make sure the referee checks Lemma 6 against [3]. If the theorem applies as stated, accept; if not, the author needs to supply the verification or modify the construction. Either way, this is publishable in a good journal after a routine check.

Referee Report

2 major / 4 minor

Summary. The paper claims a negative answer to an Erdős–Graham question: for all large N there exists A⊆{1,...,N} of size >cN such that no two distinct a,b∈A have (1/a+1/b)/2 equal to a unit fraction, equivalently a+b∤2ab. The construction is explicit: A_N consists of all a≤N for which no b≤N with Ω(b)≤Ω(a) satisfies a+b|2ab. The proof defines a set S of numbers with no prime factor <L and controlled Ω(a,x). Lemma 4 gives a lower bound |S|≫N∏_{p<L}(1−1/p). Lemma 2 reduces the divisibility a+b|2ab to u(u+v)|2a with u,v coprime. Lemma 5 bounds the number of a∈S divisible by a fixed u(u+v). The key Lemma 6 bounds a dyadic average of g1(u)g2(v)g3(u+v) by invoking [3, Theorem 3.1], yielding X^2(log X)^{−7/4}(log L)^{−3/4}. Lemma 7 uses this to show the average number of bad pairs is <1/2, so |A_N|>|S|/2>cN. The paper also notes consequences for sets of unit fractions without nontrivial three-term arithmetic progressions.

Significance. If the proof is correct, this is a substantial result: it refutes the density-zero conjecture of Erdős and Graham and gives the first positive-density construction of unit fractions with no two members whose average is a unit fraction. The explicit nature of A_N and the clean reduction to a dyadic sum of multiplicative functions are strengths. The main theorem is conditional on a correct and uniform application of the external theorem [3, Theorem 3.1]; because that application is not fully documented, the significance cannot be fully assessed until Lemma 6 is verified.

major comments (2)

[Section 1, Lemma 6] This lemma is the sole source of the dyadic bound X^2(log X)^{-7/4}(log L)^{-3/4} used in Lemma 7. The proof is a direct invocation of [3, Theorem 3.1], but the theorem is not stated and its hypotheses are not verified. Two specific points: (i) the polynomial Q=X1X2(X1+X2) has rho_Q^+(2)=4=p^2, i.e. a fixed prime divisor; the displayed singular-series product starts at 3<p, so the theorem as normally stated for admissible Q is being modified without comment; (ii) F depends on L through the factors g1,g3, and the estimate E_R<<(log X)^{5/4}(log L)^{-3/4} requires that these L-dependent local factors do not introduce extra powers of log X. Please quote [3, Theorem 3.1], verify that it permits rho_Q(p)=p^t at p=2 and an L-dependent F, or replace Lemma 6 with a self-contained proof. Without this, Lemma 7's convergence argument has no foundation.
[Section 1, Lemma 6, E_R notation] The symbol E_R is used with arguments x1+x2, x1x2, and 4(1+delta^{-1})X^2 in the same proof, with no definition. Since the estimated power is logarithmic, this may be harmless, but the reader cannot check whether the bound at the final argument is the same as the bound derived for O(X). State the definition of E_R from [3] and give the estimate at the argument actually required by Theorem 3.1.

minor comments (4)

[Lemma 4] The sentence 'For ϵ=ϵ/3 so that (1 + ϵ2 <(1 +ϵ)' is garbled; it should read something like 'Put epsilon' = epsilon/3, so that (1+epsilon')^2 < 1+epsilon'.
[Lemma 6, local factor display] The displayed local factors have denominators '2p^{-1}' and '4p^{-1}'; as written the equality with 1+5/(4p)+O(1/p^2) is false. The intended denominators are presumably 2p-1 and 4p-1.
[Lemma 7, equation (3)] The replacement of (log(3N/(u(u+v))))^{-3/4} by (log(3N/X^2))^{-3/4} is not immediate, since u(u+v) may be larger than X^2 by a factor depending on delta. It is valid up to constants depending on delta, but this should be stated.
[Introduction, acknowledgments narrative] The paragraph beginning 'The author must both acknowledge...' is out of place in a mathematical paper; if retained, it should be moved to the acknowledgments section and rewritten in conventional prose.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is explicit and the lower-bound proof rests on external analytic-number-theory theorems and a legitimate counting argument.

full rationale

The derivation chain is not circular. Lemma 2 gives an exact equivalence between a+b | 2ab and the existence of coprime u,v with b=av/u and u(u+v) | 2a. Lemma 3 then converts non-membership in the explicitly defined set A_N into an existential divisibility condition involving u(u+v), without assuming the conclusion. The set S is defined independently by small-prime sieving and an Ω(a,x) growth condition, and Lemma 4 counts S using a standard external multiplicative-functions theorem ([9, Theorem III.3.5]); no parameter is fitted to force the target density. Lemma 5 bounds the number of a∈S divisible by a fixed u(u+v) using another application of [9], giving an explicit uniform estimate. Lemma 6 invokes [3, Theorem 3.1], an external result by de la Bretèche and Tenenbaum, to bound the dyadic sum of g1(u)g2(v)g3(u+v); although the application may require checking hypotheses (a correctness risk, not circularity), the cited theorem is independent of the present paper's target and is not derived from A_N. Lemma 7 combines these bounds with the elementary fact log 4 < 3/2 and chooses ε small and L,N large; these are existential choices, not fitted predictions. Theorem 1(1) is a tautology from the definition of A_N, but that is by construction and not a circular derivation of the main claim; the substantive claim is Theorem 1(2), which is obtained by the counting argument. There are no self-citations, no imported uniqueness theorems, no ansatz smuggled via citation, and no renaming of a known result.

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

The paper introduces no empirical free parameters and postulates no new mathematical entities. The three auxiliary proof constants ε, δ, L are existential and uncomputed; the central estimates rely on two standard analytic-number-theory theorems plus elementary sieve density facts. No circularity from author self-citations is present.

free parameters (3)

ε (auxiliary proof constant) = not computed; any sufficiently small positive value with (1+ε) log 4 < 3/2
Controls the allowed excess of prime factors in S. Chosen by hand in the proof; no data fitting.
δ (auxiliary proof constant) = not computed; any fixed value in (0,1)
Sets the lower end of the interval [δN,N] where S lives and scales the v≤uδ^{-1} constraint. Existential choice only.
L (small-prime sieve cutoff) = not computed; sufficiently large depending on ε,δ
Excludes primes <L in the construction so that bad-pair density can be made arbitrarily small. Chosen existentially; no explicit value given.

assumptions (3)

standard math [9, Theorem III.3.5] bounds sums of nonnegative multiplicative functions with f(p) log p ≪ y and f(p^ν) log(p^ν)/p^ν summable
Used in Lemma 4 to estimate counts of a with Ω(a,x) overshooting, and in Lemma 5 for the 4^{-Ω(d,u+v)} sum.
standard math [3, Theorem 3.1] mean-value estimate for multiplicative functions on binary forms with k=3, t=2 and Q=X1 X2 (X1+X2)
Provides the dyadic sum bound in Lemma 6; the key external estimate in the paper.
standard math Asymptotic density ∏_{p<L}(1-p^{-1}) for numbers in [δN,N] free of primes <L
Used at the start of Lemma 4; standard inclusion-exclusion/Mertens estimate.

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Cite this review

Pith. "Pith review of Sets of unit fractions without two members whose average is a unit fraction." pith.science (2026). https://pith.science/paper/EPFL7DBL

@misc{pith2026260715419,
  author       = {Pith},
  title        = {Pith review of: Sets of unit fractions without two members whose average is a unit fraction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EPFL7DBL}},
  note         = {Machine review of arXiv:2607.15419}
}

abstract

We show that there is a constant $c>0$ such that, for all sufficiently large $N$, there is a subset $A \subseteq \{1,\dots,N\}$ of size $>cN$ such that for any two distinct elements $a,b$ in $A$, the average of $\frac{1}{a}$ and $\frac{1}{b}$ is not a unit fraction, negatively answering a question of Erd\H{o}s and Graham. This also gives the best known lower bounds on the maximum size of a set of unit fractions without non-trivial three-term arithmetic progressions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 1 linked inside Pith

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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Mean values of arithmetic functions and application to sums of powers

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Linear correlations of multiplicative functions.Proceedings of the London Mathematical Society, 121(2):372–425, April 2020

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1992