{"id":"ff0e9231-8caa-45da-a00c-b0aa939032ab","arxiv_id":"2607.15419","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"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.","lead":"An explicit construction shows that a positive proportion of the integers up to N have no pair whose reciprocals average to a unit fraction. This resolves an Erdős–Graham question in the negative and gives the strongest known lower bounds for unit-fraction sets without non-trivial three-term arithmetic progressions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6's dyadic bound rests on an unverified application of [3, Theorem 3.1] to Q=X1X2(X1+X2), which has a fixed prime divisor at p=2, and to an L-dependent F; the manuscript never checks the theorem's hypotheses.","rationale":"The central claim is the density bound |A_N|>cN. The only place this can fail within the proof is Lemma 7's estimate that the average number of bad pairs is <1/2; that estimate is obtained by applying Lemma 6, whose proof is a single external-theorem citation. The other parts (Lemmas 2, 3, 4, 5) are either elementary or have standard references, and I found no internal inconsistency in them: the Ω comparison in part (1) is correct, Lemma 4's large-sieve argument works with the stated ε' convention, and Lemma 5's Euler-product manipulation is sound up to standard O(p^-2) corrections. The genuine soft spot is Lemma 6: the polynomial Q has a local obstruction at p=2, and the function F depends on L, so the hypotheses of [3, Theorem 3.1] are not visibly satisfied. The reader identified exactly this as the weakest assumption; I agree. Because the concern is a verification gap rather than a demonstrated contradiction, the paper should be accepted only conditionally on the external theorem applying; hence verdict CONDITIONAL. The concrete check of the theorem's hypotheses and the p=2 local factor will settle it.","tokens_in":10051,"tokens_out":44226,"duration_ms":362622,"concrete_test":"Obtain the precise statement of [3, Theorem 3.1] and check its hypotheses for this Q and F: (i) does the theorem require ρ_Q^+(p)<p^t for every prime, or does it allow p=2 with ρ_Q^+(2)=4 and absorb the obstruction into E_R? (ii) does F belong to M_3(1,1,ε) with the stated submultiplicativity and local factors? Then recompute the local factors at p=2 and p∈(2,L) against the definition of E_R in [3, §2]; if the p=2 factor contributes a power of log X rather than O(1), the Euler-product estimate in Lemma 6 is wrong.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 1, Lemma 6, is the sole source of the dyadic bound X^2(log X)^{-7/4}(log L)^{-3/4} that makes Lemma 7's average-bad-pairs estimate <1/2. The proof is a direct invocation of [3, Theorem 3.1] with k=3, t=2, Q1=X1, Q2=X2, Q3=X1+X2, A=B=1. Two hypotheses are asserted, not verified. First, Q=X1X2(X1+X2) has a fixed prime divisor at p=2: ρ_Q^+(2)=4=2^2, so the displayed singular-series factor ∏_{3<p<2X}(1-(3p-2)/p^2) and the E_R factorization require the theorem to permit ρ_Q^+(p)=p^t for p=2, or to absorb the obstruction into E_R; if the theorem requires ρ_Q^+(p)<p^t, the application fails. Second, F(a1,a2,a3)=g1(a1)g2(a2)g3(a3)·1_{(a1,a2)=1} is L-dependent and has local factors at p=2 and p∈(2,L) that are only bounded in the manuscript; the claimed E_R≪(log X)^{5/4}(log L)^{-3/4} depends on these factors not contributing an extra power of log X. If either hypothesis fails, Lemma 6's exponent -7/4 changes, and Lemma 7's convergence argument (based on log 4<3/2) does not recover.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10436,"tokens_out":29097,"duration_ms":265769,"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":[{"comment":"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":"Section 1, Lemma 6"},{"comment":"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.","section":"Section 1, Lemma 6, E_R notation"}],"minor_comments":[{"comment":"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'.","section":"Lemma 4"},{"comment":"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.","section":"Lemma 6, local factor display"},{"comment":"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.","section":"Lemma 7, equation (3)"},{"comment":"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.","section":"Introduction, acknowledgments narrative"}],"recommendation":"major_revision","confidential_remarks":"The result, if correct, is publishable in a top journal. My main concern is the black-box Lemma 6: I could not verify from the manuscript that [3, Theorem 3.1] applies to Q=X1X2(X1+X2), especially with the fixed prime divisor at p=2 and the L-dependent F. Please ask the author to supply the exact theorem statement and a verification of its hypotheses. I am not recommending rejection, because the gap is local and plausibly fixable, but it is load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nI 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.","headline":"A clever positive-density construction that settles an Erdős–Graham question, pending verification of one external dyadic estimate — worth refereeing.","tokens_in":10952,"tokens_out":4234,"would_cite":true,"duration_ms":40114,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B05","11N25","11N37"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["unit fractions","Erdős–Graham problem","averages of unit fractions","prime factors counted with multiplicity","multiplicative functions","mean value theorems","three-term arithmetic progressions","positive density"],"falsifier":"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.","tokens_in":9900,"feed_emoji":"🔢","tokens_out":6762,"duration_ms":61945,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dense set exists with no two reciprocals averaging to a reciprocal","feed_subtitle":"A simple construction gives linearly many numbers up to N; their unit fractions never have a unit-fraction average.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Positive density set where no two reciprocals average to a reciprocal","Dense subset of integers avoids reciprocal-averaging pairs","Erdős–Graham question answered: dense set with forbidden averages","Unit fractions: dense set with no pair averaging to a unit fraction","No two reciprocals average to a reciprocal in a dense set"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Positive density set where no two reciprocals average to a reciprocal","Dense subset of integers avoids reciprocal-averaging pairs","Erdős–Graham question answered: dense set with forbidden averages","Unit fractions: dense set with no pair averaging to a unit fraction","No two reciprocals average to a reciprocal in a dense set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":971,"prompt_tokens":600,"completion_tokens":371,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":344,"completion_tokens_details":{"reasoning_tokens":297}},"tokens_in":344,"tokens_out":371,"duration_ms":3513,"temperature":1.0,"reasoning_tokens":297,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:27:17.846291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}