{"id":"3e3f5130-7326-4900-9123-6da526472b9c","arxiv_id":"2502.09539","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Erdős's integer dilation approximation problem is resolved: sets with positive logarithmic density always contain distinct α, β with |nα−β| arbitrarily small.","lead":"A long-open problem of Erdős from 1948 is solved: any set of real numbers with positive logarithmic density contains distinct numbers α and β such that some integer multiple of α is arbitrarily close to β. The proof adapts the GCD graph machinery from the Duffin-Schaeffer conjecture and adds a new balance from Behrend's estimate.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central GCD-graph machinery is inherited from the unpublished preprint [22] via assertions of 'minimal changes'; this is a genuine verification gap, but no concrete error was found, so the ACCEPT verdict stands unchanged.","rationale":"The reader's weakest assumption correctly identified the dependence on adaptations from the unpublished preprint [22] as the most fragile point. My independent pass through the manuscript found no concrete contradiction or incorrect computation in the sections that are written out: the reduction of Theorem 1 to Proposition 2.15 is coherent, the second-moment setup and the construction of A′ are internally consistent, the proof of Lemma 3.1 is detailed, and the reduction of Proposition 2.15 to Proposition 6.2 in Section 6 checks out. The remaining risk is that Propositions 7.11-7.14, and especially Lemma 9.2 and Lemma 9.4 in the rational-vertex setting, may not survive the move from integer vertices and nonnegative valuations to rational vertices and integer valuations, or that the modified quality function without the Euler factors may break the quality-increment arguments. This is exactly the risk the reader flagged. Because no specific failure was identified and because the authors are the same as in [22], the preprint's ACCEPT with moderate confidence remains appropriate. If a referee requires full self-contained proofs of the GCD-graph propositions before publication, the appropriate verdict would be CONDITIONAL, but I do not see a reason to change the reader's verdict on the current record.","tokens_in":44669,"tokens_out":30277,"duration_ms":257177,"concrete_test":"Independently transcribe the proof of Lemma 9.4 from [22, Lemma 13.2] into the rational-vertex setting, allowing k and ℓ in Z, and verify the dichotomy in the case of negative p-adic valuations: the constructed exact maximal subgraph must satisfy q(G') ≥ M^{1_{f'(p)≠g'(p)}} q(G) and R(G') ⊆ R(G)\\setminus{p}. As an explicit benchmark, run the dichotomy on G = (λ, V=W={1/p,1}, complete bipartite edges) with p > C2 and k = -1; if neither q(G_{p^{-1},p^{-1}}) ≥ M q(G) nor the complementary edge bound μ(E \\setminus E_{p^{-1},p^{-1}}) ≤ μ(E)/(4p^{1+τ/4}) holds, the 'minimal changes' claim is falsified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the key estimate Proposition 2.15 rests on Proposition 6.2, whose proof in Section 8 depends on the GCD-graph Propositions 7.11-7.14. These propositions are not proved self-containedly: Lemmas 9.2, 9.4, 9.5 and 9.6 are stated as adaptations of [22, Lemmas 11.3, 13.2, 14.1, 14.2], with the only differences being that vertices are rational, p-adic valuations k and ℓ are allowed to be negative, and the quality function omits the Euler factors of [22]. The deductions of Propositions 7.12 and 7.14 are likewise referred back to [22], and Lemma 11.1 invokes [22, Lemma 15.1]. If any of these adaptations is incorrect, the chain that bounds E3 by O(J) in Section 2.6 can fail, and with it Theorem 1. The paper itself repeatedly flags these as 'minimal changes' (Sections 9.1, 9.2, 10, 11) rather than proving them, so the load-bearing premise is exactly the correctness of the rational-vertex, modified-quality transfer from [22]. I did not find an explicit mathematical error in the portions of the proof that are written out, and no internal inconsistency surfaced in Sections 2-6 and 8; the risk is concentrated in the unpublished-preprint dependency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Erdős's 1948 integer dilation approximation problem under the logarithmic-density condition (1.3): if A ⊂ R>0 is discrete with limsup_{x→∞} (1/log x) Σ_{α∈A∩[1,x]} 1/α > 0, then for every ε>0 there are infinitely many distinct α,β∈A and a positive integer n with |nα−β|<ε. The proof uses a second-moment method over the rough-number events N_α, analyzes their correlations through a new bracket [α,β], and reduces the hard correlation case to a bound on weighted bipartite GCD graphs whose vertices are rational numbers. The key estimate is Proposition 2.15, which is reduced to Proposition 6.2 and then to a series of GCD graph propositions (7.11–7.14). The paper is well organized and contains a self-contained proof of a refinement of Behrend's theorem (Theorem 4.1) and a substantial portion of the GCD graph arguments, but the proof of the central propositions relies on adaptations from the unpublished preprint [22] that are stated but not carried out.","tokens_in":45029,"tokens_out":4413,"duration_ms":42483,"significance":"If correct, this is a major advance: it resolves a problem Erdős posed in 1948 under condition (1.3) and introduces a novel combination of the GCD graph machinery with rational vertices, negative p-adic valuations, and a Behrend-type sieve. The paper gives a clear structure-versus-randomness heuristic, explicit reductions, and no evident circularity. The constants θ=2.001 and M=e^4 are chosen to make the iterative argument work and appear to be legitimate. However, the proof's load-bearing part—the GCD graph propositions—is not self-contained: Propositions 7.11–7.14 depend on Lemmas 9.2, 9.4, 9.5, 9.6, and 11.1, which are asserted to follow from [22] with 'minimal changes.' This is a genuine verification gap that should be addressed before the paper can be accepted as a complete proof.","major_comments":[{"comment":"The proof of Proposition 6.2, and hence of Theorem 1, relies on Propositions 7.11–7.14, whose proofs are not given. Instead, Sections 9–11 state that certain lemmas are direct adaptations of results in the unpublished preprint [22], with only 'minimal changes' (e.g., §9.1, §9.2, and the context of Lemma 11.1 in Section 11). These lemmas are load-bearing: Lemma 9.4 is used in Proposition 7.12, Lemma 9.6 in Proposition 7.11, Lemmas 9.2/9.3 in Proposition 7.13, and Lemma 11.1 in Proposition 7.14. Because the changes involve rational vertices, negative p-adic valuations, and a modified quality function without the Euler factors of [22], the correctness of these adaptations is not verifiable from the text. The manuscript should either provide complete proofs of the adapted lemmas, or a precise line-by-line correspondence with [22] displaying all modifications and verifying them. Without this, the central estimate (2.23) is not fully established.","section":"Sections 9-11"}],"minor_comments":[{"comment":"The notation '3ω' in the definition of η (and in the proof) should be '3^{ω}'; the proof clearly uses the exponential form 3^{ω(...)} in bounding S. If the printed version indeed uses 3ω, it is a typo that makes the estimate dimensionally wrong.","section":"Section 3.2"},{"comment":"The terminology 'negatively correlated' is formally incorrect, as the condition is asymptotic non-positive correlation; the authors acknowledge this in the footnote, but the main text would benefit from a brief remark or a slightly different name.","section":"Section 2.2"},{"comment":"Reference [22] is cited as 'Duke Math. J., to appear' but the proof depends essentially on it; the reference should be updated to the published version, or its availability should be confirmed, to allow readers to verify the minimal changes.","section":"References"},{"comment":"The sentence 'the remaining pairs are very few' is vague; it would be clearer to state explicitly that they are handled by Proposition 2.15, whose statement follows.","section":"Section 2.5"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the heavy reliance on the unpublished preprint [22] with only 'minimal changes' assertions. If the editors obtain written assurance from the authors that [22] will appear promptly and that the adaptations have been independently checked, the paper may be acceptable after that. Otherwise, the authors should supply the missing proofs in an appendix. This is a verification gap, not a discovered error; no circularity or concrete mathematical mistake was found in the written portions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a genuine resolution of a sixty-plus-year-old problem, and the main body of the proof is in good shape. The theorem — positive logarithmic density forces some |nα − β| < ε for distinct α, β in the set — is new; Haight only handled the all-irrational-ratio case. The high-level strategy is clear: restrict to rough multiples, use the second moment method, show irrational ratios are uncorrelated, then use GCD graphs to control rational-ratio correlations. The two genuinely new moves are the adaptation of the GCD-graph machinery to rational vertices (tracking numerator and denominator GCDs together) and the use of a generalized Behrend estimate to balance the extra summation over the possible fixed divisors A and B. That balance is the kind of thing you either see or you don't, and here it is spelled out.\n\nWhat the paper does well: the reduction to Proposition 2.15 is explicit and the intermediate lemmas (correlation estimates, the construction of the sets A′_j, the generalized Behrend theorem, the deduction of the rational-set Proposition 6.2) are proved carefully in the text. The writing is unusually honest about the structure of the argument, including the limitation that the final sections do not yield the stronger quantitative estimate (1.6).\n\nThe soft spot is exactly what the stress test flags: Propositions 7.11–7.14, which carry the weight in Section 8, are proved using Lemmas 9.2, 9.4, 9.5, 9.6 and 11.1, and for all of those the paper says 'the argument of [22] goes through with minimal changes' (or words to that effect). The changes are real — rational vertices, negative p-adic valuations, a modified quality function — and the paper does not include the transfer proofs. Since [22] is a preprint (even if by largely overlapping authors), a referee cannot fully verify those assertions without doing substantial work. I did not find a concrete error in the written portions, and the dependence is flagged rather than hidden, but it is still a load-bearing reliance on an unpublished source. If any of those adaptations is wrong, the E_3 bound fails and with it Theorem 1. That said, the surrounding arguments in Sections 2–6 and 8 are coherent, and the machinery from [21] is established enough that the 'minimal changes' claim is plausible.\n\nWho this is for: people working in Diophantine approximation, primitive sets, and GCD-graph methods. The paper deserves a serious referee — it is important, new, and largely self-contained except for the [22] dependence. My recommendation: send it to a strong number theory journal, and instruct the referee to put real effort into checking Section 9–11 against [22], or ask the authors to supply a fuller appendix. The dependence is unpublishable in its current form only if the referee finds a concrete break; otherwise it should be accepted with a request for additional verification of those transfers.","headline":"This paper resolves Erdős's 1948 dilation problem under logarithmic density condition (1.3) with a convincing proof, though a few key lemmas are inherited from an unpublished preprint via 'minimal changes' assertions.","tokens_in":45497,"tokens_out":1370,"would_cite":true,"duration_ms":17160,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J25","11B83","11A05","05C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Positive reciprocal density guarantees an integer dilation pair","keywords":["integer dilation approximation","GCD graphs","primitive sets","Diophantine approximation","second moment method","logarithmic density","rough numbers","structure versus randomness"],"falsifier":"Verify Proposition 2.15 numerically on finite, 1-spaced sets B of rational numbers with controlled heights and bracket sizes: compute λ of the pairs with H(α/β)≤$x^{3}$, y<[α,β]≤2y, and L(α/β;z)>1, and check whether the bound λ({(α,β)∈B×B:...}) ≪ y $e^{{-z}}$ (log x)^2 holds for all admissible x,y,z. A single counterexample would directly refute the key reduction and hence Theorem 1.","tokens_in":44497,"feed_emoji":"🔢","tokens_out":7710,"duration_ms":76268,"temperature":0.7,"pith_summary":"A discrete set of positive real numbers with positive reciprocal logarithmic density must contain two distinct elements α and β for which some positive integer n makes |nα−β|<ε, for every ε>0. The paper proves this 1948 Diophantine approximation problem under condition (1.3), and the proof actually yields infinitely many such pairs. The significance is that a purely measure-theoretic divergence condition forces a number-theoretic approximation phenomenon, even when the set is discrete and has no accumulation points. The argument combines a second moment method over intervals with a structural dichotomy: sets that avoid close integer dilations must either be mostly random, where averaging works, or contain a large primitive-like structured part, where a refinement of a classical primitive-set estimate supplies the needed saving.","feed_headline":"Positive reciprocal density forces close integer dilations","feed_subtitle":"Discrete sets with positive reciprocal logarithmic density always contain two nearly aligned integer dilations.","key_machinery":"The load-bearing mechanism is the GCD-graph machine, imported from work on a related Diophantine approximation problem and adapted to rational vertices. Alongside it, the proof introduces the bracket [α,β]=H(α/β)/max{α,β}, which measures the height of a rational ratio, and replaces the intervals Mα by thinner events Nα whose multipliers n are α-rough, meaning all prime factors exceed α. These choices make disjointness and negative correlation easy: when [α,β]≤1 the events Nα and Nβ do not overlap at all, and when [α,β] is large the correlation is bounded by sieving estimates. The remaining small-bracket pairs are shown, via maximal GCD subgraphs and a quality function q(G), to concentrate in a structured subgraph, to which a refined primitive-set estimate (Theorem 4.1) is applied. That refinement, itself a generalization of a classical primitive-set result, supplies exactly the savings needed to balance the extra summation over possible fixed divisors, a feature the paper singles out as new.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1: for any discrete A⊂R>0 with limsup_{x→∞} (1/log x) ∑_{α∈A∩[1,x]} 1/α > 0 and any ε>0, there exist distinct α,β∈A and n∈N with |nα−β|<ε. Iterating the theorem after deleting each found pair gives infinitely many such pairs. The proof proceeds by contradiction, assuming that |nα−β|≥1 for all distinct α,β and all n∈N. From that assumption the paper derives lim_{T→∞} (1/T)∑_{α∈A∩[1,T]}1 = 0, which contradicts the divergence of the reciprocal sums. This density-zero conclusion is exactly what resolves the problem; no quantitative rate such as (1.6) is obtained, and the authors explicitly describe the proof as soft.","pith_inferences":["Because the final saving is exactly balanced by the extra summation over fixed divisors, a quantitative analogue with a rate would likely require a new mechanism to break that balance; the paper only obtains the qualitative o(log x) conclusion.","The same GCD-graph dichotomy may be adaptable to prove stronger distribution statements about the ratios of elements of A, for example that the ratios cannot all stay away from the integers in a weighted second-moment sense.","The square-free-numerator case singled out in the paper is a natural intermediate target: with square-free numerators the added denominator-only iteration could yield the stronger estimate (1.5), and testing that case would isolate how much of the full theorem depends on the delicate final balancing step."],"forward_implications":["Every set A satisfying condition (1.3) contains infinitely many distinct pairs (α,β) with |nα−β|<ε for each ε>0, obtained by repeatedly deleting already-found pairs.","The 1948 problem is settled in its contrapositive form under the logarithmic divergence condition: no discrete set with positive reciprocal logarithmic density is free of close integer dilations.","The random-versus-structured dichotomy becomes a usable proof architecture: either the second moment over α-rough events succeeds directly, or a large structured subset supports a Behrend-type saving.","The proof is deliberately non-quantitative; it shows the counting function is o(log x) without a rate, and the paper states that quantitative estimates like (1.6) are not proved."],"supporting_citations":[{"why":"It states the original 1948 problem and lists the two divergence conditions, one of which, (1.3), is the paper's target.","marker":"[5]"},{"why":"It supplies the second-moment method for interval events and the irrational-ratio correlation lemma that the paper refines and extends.","marker":"[17]"},{"why":"It introduces the GCD-graph machinery and the maximality and quality arguments that the paper adapts to rational vertices.","marker":"[21]"},{"why":"It is the companion preprint whose propositions, adapted with minimal changes in Sections 9 through 11, provide the crucial GCD-graph lemmas.","marker":"[22]"},{"why":"It supplies the classical primitive-set estimate that the paper generalizes into Theorem 4.1 and uses as the final balancing saving.","marker":"[2]"},{"why":"It provides the rough-number construction for the events Nα and the logarithmic-density estimate for primitive sets that motivates the structured case.","marker":"[4]"},{"why":"It contributes the variant of GCD-graph iteration, involving only partial determination of numerator divisors, that the paper borrows to gain the needed Euler factors.","marker":"[19]"}],"fun_headline_variants":["Positive reciprocal density yields close integer dilations","Erdős's 1948 dilation problem resolved by reciprocal density","Reciprocal density positivity implies near integer dilations","Near integer dilations inevitable for positive reciprocal density","Close integer dilations guaranteed by reciprocal logarithmic density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a battery of GCD-graph lemmas taken from an unpublished companion preprint, applied to rational vertices with a modified quality function, with the paper asserting that the adaptations require only minimal changes; if any of those adaptations is not actually valid, the central claim is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Positive reciprocal density yields close integer dilations","Erdős's 1948 dilation problem resolved by reciprocal density","Reciprocal density positivity implies near integer dilations","Near integer dilations inevitable for positive reciprocal density","Close integer dilations guaranteed by reciprocal logarithmic density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00071,"raw_usage":{"total_tokens":3158,"prompt_tokens":871,"completion_tokens":2287,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2212}},"tokens_in":487,"tokens_out":2287,"duration_ms":17133,"temperature":1.0,"reasoning_tokens":2212,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:05:51.910483+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify Proposition 2.15 numerically on finite, 1-spaced sets B of rational numbers with controlled heights and bracket sizes: compute λ of the pairs with H(α/β)≤$x^{3}$, y<[α,β]≤2y, and L(α/β;z)>1, and check whether the bound λ({(α,β)∈B×B:...}) ≪ y $e^{{-z}}$ (log x)^2 holds for all admissible x,y,z. A single counterexample would directly refute the key reduction and hence Theorem 1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It states the original 1948 problem and lists the two divergence conditions, one of which, (1.3), is the paper's target."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the second-moment method for interval events and the irrational-ratio correlation lemma that the paper refines and extends."},{"cited_title":"Koukoulopoulos and J","cited_arxiv_id":null,"evidence_quote":"It introduces the GCD-graph machinery and the maximality and quality arguments that the paper adapts to rational vertices."},{"cited_title":"Koukoulopoulos, J","cited_arxiv_id":null,"evidence_quote":"It is the companion preprint whose propositions, adapted with minimal changes in Sections 9 through 11, provide the crucial GCD-graph lemmas."},{"cited_title":"Behrend, On sequences of numbers not divisible by another","cited_arxiv_id":null,"evidence_quote":"It supplies the classical primitive-set estimate that the paper generalizes into Theorem 4.1 and uses as the final balancing saving."},{"cited_title":"Erd˝ os,Note on sequences of integers no one of which is divisible by a ny other","cited_arxiv_id":null,"evidence_quote":"It provides the rough-number construction for the events Nα and the logarithmic-density estimate for primitive sets that motivates the structured case."},{"cited_title":"Proving the Duffin-Schaeffer conjecture without GCD graphs","cited_arxiv_id":"2404.15123","evidence_quote":"It contributes the variant of GCD-graph iteration, involving only partial determination of numerator divisors, that the paper borrows to gain the needed Euler factors."}],"review_version":1}