{"id":"370e944f-145c-471f-bb26-489bbf82f6f6","arxiv_id":"2506.04187","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The moving-target Khintchine conjecture holds under an extra-divergence condition on psi and whenever the target centers lie in a finite set, yielding a monochromatic-denominator corollary.","lead":"This paper proves two partial cases of a conjecture that Khintchine's theorem keeps working when the target point moves with the denominator: when the approximating function diverges with an extra logarithmic factor, and when the target can only take finitely many values. A byproduct gives rational approximations whose denominators all have one color in any finite coloring.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 18's intersection-counting threshold appears to be the wrong weight, so the proof of the QIA estimate used for Theorem 3 has a gap as written.","rationale":"The reader's verdict accepted the paper and flagged Proposition 18 as a black-box input inherited from Schmidt. My stress-test agrees that Proposition 18 is the load-bearing estimate for Theorem 3, but finds a concrete internal inconsistency in its proof as printed: the counting function M(q,r) uses the threshold 2r psi(q), whereas the covering inequality and all subsequent estimates require the threshold 2q psi(r) (or the exact threshold r psi(q)+q psi(r)). For decreasing psi and r<q these thresholds differ by a factor that can be as large as q/r, so the displayed M undercounts intersections badly. A simple numerical example confirms the issue. The theorem may well be true and the proof may be repairable by replacing the threshold, but as written the proof of Proposition 18 does not establish the QIA bound on which Theorem 3 depends. I did not find a comparable issue in the proof of Theorem 1: the application of Proposition 8, Lemma 11, and the transfer argument are coherent once the intended iterated-logarithm condition is read as an extra-divergence condition with the log factors in the denominator. The finite-center theorem is the one whose written proof currently has a gap, so a conditional acceptance pending correction of the intersection-counting threshold is the appropriate outcome.","tokens_in":18647,"tokens_out":25369,"duration_ms":244261,"concrete_test":"Re-derive the intersection condition and test the printed inequality numerically for q=100, r=10, gamma=0, psi(n)=1/n: compute m(A'_q cap A'_r) and compare it with (2 psi(q)/q) M(q,r) using the printed threshold 2r psi(q). If the printed M gives zero while the intersection measure is positive, the proof as written fails. Then replace the threshold by 2q psi(r) (or by r psi(q)+q psi(r)) and check that the inequalities M(q,r) << q psi(r), M(q,r) << gcd(q,r), and M(q,r) <= N_{1/5}(q,r) used in the three terms of Proposition 18 all hold with this corrected definition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Proposition 18, for r<q the sets A'_q and A'_r are unions, over a in S(q) and b in S(r), of intervals of radii psi(q)/q and psi(r)/r. An intersection contributes only when |(a+gamma)r-(b+gamma)q| < r psi(q)+q psi(r). The paper defines M(q,r) with the smaller threshold 2r psi(q) and then asserts m(A'_q cap A'_r) <= (2 psi(q)/q) M(q,r). This inequality is false in general: take gamma=0, psi(n)=1/n, q=100, r=10. Then r psi(q)+q psi(r)=0.1+10=10.1, while 2r psi(q)=0.2, so M counts almost none of the intersecting pairs. The later estimates in Proposition 18, which use conditions such as q psi(r) >= gcd(q,r), are consistent only if the threshold is 2q psi(r) (or the exact sum r psi(q)+q psi(r)). Thus the quasi-independence on average statement on which Theorem 3 is built is not proved as written. This looks like a fixable typo rather than a fatal flaw, but the proof of Theorem 3 is incomplete until the threshold is corrected and the argument re-verified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the one-dimensional inhomogeneous Khintchine theorem with a moving target. Defining W(ψ,γ) = {α∈[0,1]: ||qα−γ_q||<ψ(q) infinitely often}, the authors prove two partial results toward the Hauke–Ramírez Conjecture 1: Theorem 1 gives full measure under an extra-divergence condition, and Theorem 3 gives full measure when the centers γ_q lie in a finite set, with Corollary 5 on monochromatic denominators. The proof of Theorem 1 combines an overlap estimate (Lemma 15), a quasi-independence criterion (Proposition 8), and divisor-sum estimates via Kac's theorem (Lemma 11). The proof of Theorem 3 uses Schmidt's fixed-center QIA estimate (Proposition 18) and a pigeonhole argument to pass to one color class, then applies a Borel–Cantelli criterion from Beresnevich–Hauke–Velani.","tokens_in":18900,"tokens_out":34756,"duration_ms":318000,"significance":"If the main theorems are corrected as indicated below, the paper would constitute a genuine advance on a conjecture from Hauke and Ramírez, and Corollary 5 is a new monochromatic-denominator inhomogeneous Khintchine theorem. The paper is written carefully, uses appropriate external benchmarks (Szüsz, Schmidt, Sprindžuk, BDV, BHV, Kac), and its abstract QIA machinery is a useful framework. The main proofs are not machine-checked, but they are detailed enough to review line by line. Two load-bearing points currently need repair: the printed extra-divergence hypothesis of Theorem 1 does not match the proof, and the QIA estimate in Proposition 18 uses an incorrect intersection threshold. Both appear fixable within the manuscript's framework.","major_comments":[{"comment":"The hypothesis of Theorem 1 is printed as ∑ ψ(q)√(log q (log log q)...(log...log (k iterates) q)^{1+ε}) = ∞. Since the radical is at least 1 for all sufficiently large q, this condition is implied by the plain divergence ∑ψ(q)=∞; it is therefore not an 'extra divergence' assumption and, as stated, the theorem would claim the full Conjecture 1. The proof in §4.1 instead uses the condition ∑ψ(q)/(f(log log q)√log q)=∞ with f(x)=x(log x)...(log...log (k−2 iterates) x)^{1+ε}, i.e. the square root appears in the denominator. These are different hypotheses. The statement in the abstract, the introduction, and the proof must be reconciled (most likely by placing the radical in the denominator), and Corollary 2 and the 'in particular' remark should be checked against the corrected condition.","section":"§1.1 / §4.1 (Theorem 1)"},{"comment":"The proof of Proposition 18 defines M(q,r) with the threshold |(a+γ)r−(b+γ)q| < 2rψ(q) and then asserts m(A'_q∩A'_r) ≤ (2ψ(q)/q)M(q,r). The actual nonempty-intersection threshold is rψ(q)+qψ(r). The asserted inequality is false in general: for γ=0, ψ(n)=1/n, q=100, r=10, one has rψ(q)+qψ(r)=10.1 and, e.g., the pair a=9, b=1 gives an intersection, while 2rψ(q)=0.2 and M(100,10)=0 for the reduced sets S(100), S(10). Thus the QIA estimate ∑_{q,r≤Q} m(A'_q∩A'_r) ≪ Ψ(Q)², which is the load-bearing input for Theorem 3, is not proved as written. The fix is to replace the threshold by rψ(q)+qψ(r) (or by a comparable upper bound such as 2qψ(r)); the subsequent estimates in the proof appear consistent with this correction because rψ(q) ≤ qψ(r), but the proposition should be re-verified with the corrected threshold.","section":"§5, Proposition 18"}],"minor_comments":[{"comment":"In the verification of condition (8), the displayed chain m(A'_{q,k}∩I) ≤ (ψ(q)/q)#(S(q)/q∩I) misses the factor 2 coming from the length of each component interval; as written it is false, e.g. when I is one of the component intervals of A'_{q,k}. Replacing ψ(q)/q by 2ψ(q)/q repairs the chain and still gives (8).","section":"§6, proof of Theorem 3"},{"comment":"In the application of Proposition 8, η(q,r)=gcd(q,r)/q can be smaller than 1, while Proposition 8 states η:N²→[1,∞). This can be fixed by replacing η with max{η,1} or by extending the statement to nonnegative η, but the mismatch should be addressed.","section":"§4.1"},{"comment":"The iterated-logarithm notation in Theorem 1, Corollary 2, and the proof is ambiguous; please define the number of iterates explicitly, for example with log_k x, so that the exponents in the statement and in the definition of f(x) can be checked.","section":"§1.1 and §4.1"}],"recommendation":"major_revision","confidential_remarks":"The two main issues are local and repairable: the Theorem 1 statement appears to have a typesetting error in the extra-divergence condition, and Proposition 18's threshold error is a fixable but load-bearing gap. I would not reject; a careful revision with the corrected Theorem 1 condition and a re-verified Proposition 18 would make the paper publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is the first real progress on Hauke–Ramirez's moving-target Khintchine conjecture: they prove it under an extra-divergence condition and for finitely many centers, plus a monochromatic-denominator corollary. Second, there is a real gap in Proposition 18 as written, but it looks like a typo, not a fatal flaw.\n\nThe extra-divergence theorem is new and nontrivial: it covers psi(q)=1/q, which Sprindzuk's fast-divergence asymptotic cannot, and the proof via Yu's abstraction plus Kac's divisor CLT is clean. The finite-center theorem is also new, and the pigeonhole reduction to Schmidt's fixed-center QIA is a neat idea. The paper is honest about the literature: the conjecture from [17] is the target of the proof, not an input, and the citation pattern is not circular.\n\nThe soft spot is Proposition 18. The intersection condition for A'_q and A'_r is |(a+gamma)r - (b+gamma)q| < r psi(q) + q psi(r), but M(q,r) is defined with the smaller threshold 2r psi(q). Since psi is decreasing, r psi(q) <= q psi(r), so the true threshold is between q psi(r) and 2q psi(r). As written, the bound m(A'_q cap A'_r) <= (2 psi(q)/q) M(q,r) is false: M undercounts the intersecting pairs. The subsequent estimates are consistent with replacing 2r psi(q) by 2q psi(r); then the first term gives M << q psi(r) when q psi(r) >= gcd(q,r), and the third term works because r psi(q) <= q psi(r) lets Lemma 17 apply with delta=1/5. So the fix is straightforward and Theorem 3 should go through, but the proof is incomplete until that correction is made and verified. This is exactly the kind of thing a referee should catch and the authors should fix.\n\nThe paper ships no code; it relies on standard black boxes (Schmidt, Yu, Beresnevich–Hauke–Velani), which is normal for this area. The extra-divergence threshold with iterated logs is natural and the exposition is clear.\n\nWho is it for: people in metric Diophantine approximation and shrinking-target dynamics. It deserves a serious referee. I would send it to review, with instructions to have the authors correct the threshold in Proposition 18 and reverify the three-term split.","headline":"A genuinely new partial proof of the moving-target Khintchine conjecture, with a fixable threshold typo in the finite-center argument that a referee should catch.","tokens_in":19436,"tokens_out":4786,"would_cite":true,"duration_ms":44623,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","11K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The moving-target conjecture for Khintchine's theorem is proved in two regimes: under an extra-divergence condition on the approximation function, and when the moving centers lie in a finite set.","keywords":["Diophantine approximation","inhomogeneous approximation","shrinking targets","moving targets","limsup sets","quasi-independence on average","divisor function","Khintchine's theorem"],"falsifier":"A concrete check: fix $\\psi(q)=1/q$ and let $\\gamma_q$ be the fractional part of $\\sqrt{q}$. The theorem requires $m(W(\\psi,\\gamma))=1$. Compute the intersection sum $\\sum_{q,r\\le Q}m(A_q\\cap A_r)$ and the quasi-independence ratio used in the overlap argument; if the ratio does not stay bounded away from zero as $Q\\to\\infty$, the overlap absorption has failed for this target. Alternatively, a single explicit sequence $(\\gamma_q)$ and decreasing $\\psi$ with divergent extra-divergence sum but $m(W(\\psi,\\gamma))<1$ would refute the theorem outright.","tokens_in":1837,"feed_emoji":"🎯","tokens_out":6920,"duration_ms":206440,"temperature":0.7,"pith_summary":"The paper concerns the classical inhomogeneous Khintchine theorem, which says that for a fixed offset, almost every real number is approximated infinitely often by shifted rationals with error governed by a decreasing function, provided its sum diverges. It addresses the open case where the offset is allowed to depend on the denominator, so the target both shrinks and moves. The paper proves the moving-target conjecture for two classes: approximation functions satisfying an extra-divergence condition, effectively requiring a divergent sum after dividing by a square-root-of-log factor and iterated logarithms, and target-center sequences contained in a finite set. In both cases the set of approximated points has full Lebesgue measure. A byproduct is a finite-coloring theorem: in any finite coloring of the denominators, one color class alone supports full-measure approximation.","feed_headline":"Moving-target Khintchine proven for extra divergence, finite centers","feed_subtitle":"Two settings where the moving-target conjecture holds: extra-divergent error sums and finitely many center positions.","key_machinery":"The central object is the limsup set $W(\\psi,\\gamma)=\\{\\alpha\\in[0,1]:\\|q\\alpha-\\gamma_q\\|<\\psi(q)\\text{ infinitely often}\\}$, encoding visits of the orbit $q\\alpha\\bmod 1$ to shrinking intervals centered at $\\gamma_q$. Theorem 1 runs on an overlap estimate $m(A_q\\cap A_r)\\le 2m(A_q)m(A_r)+\\frac{\\gcd(q,r)}{q}m(A_q)$ and an abstract quasi-independence lemma that absorbs the gcd term under an extra-divergence condition. A divisor-count transfer converts the assumed divergence into divergence of $\\sum \\psi(q)/(f(\\log d(q))d(q))$, using the central limit theorem for the divisor function and the estimate $\\sum_{n\\le x}1/d(n)\\asymp x/\\sqrt{\\log x}$. Theorem 3 instead uses a fixed-center quasi-independence estimate and a pigeonhole step that selects one color class, together with an equidistribution statement for the sets of admissible shifted numerators; the full-measure conclusion then follows from a Borel–Cantelli-type full-measure criterion.","core_discovery":"On the paper's own terms, the discovery is that the moving-target conjecture holds under two additional hypotheses. The first, Theorem 1, states that if for some $\\varepsilon>0$ and $k\\ge 2$ the sum $\\sum_{q=1}^{\\infty} \\psi(q)/(\\sqrt{\\log q}\\,(\\log\\log q)\\cdots(\\log^{(k)} q)^{1+\\varepsilon})$ diverges, then $m(W(\\psi,\\gamma))=1$ for every sequence $(\\gamma_q)$ of centers. The second, Theorem 3 and Corollary 4, requires only the classical divergence $\\sum\\psi(q)=\\infty$ but constrains the centers to a finite set $\\{\\sigma_1,\\dots,\\sigma_\\ell\\}$; then there is some $k$ such that the restricted limsup set $W(1_{\\{\\gamma_q=\\sigma_k\\}}\\psi,\\sigma_k)$ has full measure. Theorem 3 also yields a finite-colorings statement: for any finite partition of $\\mathbb{N}$, some cell $\\pi$ satisfies $m(W(1_\\pi\\psi,\\gamma))=1$ for any fixed $\\gamma$. These are genuine extensions of the fixed-center theorem, not just convergence-side observations.","pith_inferences":["The extra $\\sqrt{\\log q}$ factor in the threshold is likely an artifact of the inhomogeneous gcd term, and a sharper overlap estimate could plausibly lower it to $\\sum\\psi(q)(\\log q)^{-\\varepsilon}=\\infty$, matching the analogous extra-divergence results the paper cites.","The finite-color theorem suggests a route to the full conjecture: by the paper's own reduction it suffices to prove the conjecture for centers in a countable dense set such as the rationals, so a pigeonhole or quasi-independence argument over a countable family of centers might close the problem.","The divisor-count transfer indicates the overlap term is essentially controlled by the divisor function; testing whether $\\sum \\psi(q)/d(q)=\\infty$ is already sufficient would be a direct stress test of the method."],"forward_implications":["For any decreasing $\\psi$ satisfying the extra-divergence condition, $m(W(\\psi,\\gamma))=1$ for every moving target; in particular this covers $\\psi(q)=1/q$.","If the centers $\\gamma_q$ are drawn from a finite set, the classical divergence condition $\\sum\\psi(q)=\\infty$ alone is enough, and one of the finitely many centers is responsible for full measure on its own subsequence.","In any finite coloring of the denominators, some color class $\\pi$ yields $m(W(1_\\pi\\psi,\\gamma))=1$, giving rational approximations with monochromatic denominators.","A fast-divergence asymptotic formula previously covered extremely large error sums; Theorem 1 lowers the threshold to a tractable extra-divergence condition that covers natural functions like $1/q$."],"supporting_citations":[{"why":"The fixed-center inhomogeneous Khintchine theorem that the paper sets out to generalize to moving targets.","marker":"[25]"},{"why":"Supplies the asymptotic formula for the counting function in the moving-center case, used to prove the fast-divergence proposition and as the baseline for extra divergence.","marker":"[24]"},{"why":"Provides the abstraction behind the quasi-independence lemma that absorbs the gcd overlap term under extra divergence.","marker":"[28]"},{"why":"Provides the fixed-center quasi-independence estimate and the shifted reduced-denominator sets on which the finitely-centered theorem rests.","marker":"[23]"},{"why":"Supplies the central limit theorem for the divisor function used in the divisor-count transfer.","marker":"[19]"},{"why":"Provides the estimate $\\sum_{n\\le x}1/d(n)\\asymp x/\\sqrt{\\log x}$ used in the divisor-count transfer.","marker":"[26]"},{"why":"Supplies the local full-measure criterion used after Theorem 1 establishes positive measure on every open set.","marker":"[5]"},{"why":"Supplies the full-measure quasi-independence criterion used to finish the finitely-centered theorem.","marker":"[7]"}],"fun_headline_variants":["Moving-target Khintchine proven with extra divergence","Finite centers suffice for moving-target Khintchine","Khintchine for moving targets: extra divergence or finite sets","Inhomogeneous Khintchine extends to moving centers","Extra divergence or finite centers: moving-target theorem"],"cache_read_input_tokens":21632,"weakest_assumption_plain":"The entire argument rests on being able to absorb the overlap term $\\gcd(q,r)\\psi(q)/q$ into the square of the sum of the measures; if that absorption fails at the stated divergence thresholds, the quasi-independence argument yields only a positive-measure set rather than full measure.","fun_headline_variants_meta":{"raw":{"variants":["Moving-target Khintchine proven with extra divergence","Finite centers suffice for moving-target Khintchine","Khintchine for moving targets: extra divergence or finite sets","Inhomogeneous Khintchine extends to moving centers","Extra divergence or finite centers: moving-target theorem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000901,"raw_usage":{"total_tokens":3932,"prompt_tokens":1049,"completion_tokens":2883,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":2807}},"tokens_in":665,"tokens_out":2883,"duration_ms":21792,"temperature":1.0,"reasoning_tokens":2807,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:47:28.730159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: fix $\\psi(q)=1/q$ and let $\\gamma_q$ be the fractional part of $\\sqrt{q}$. The theorem requires $m(W(\\psi,\\gamma))=1$. Compute the intersection sum $\\sum_{q,r\\le Q}m(A_q\\cap A_r)$ and the quasi-independence ratio used in the overlap argument; if the ratio does not stay bounded away from zero as $Q\\to\\infty$, the overlap absorption has failed for this target. Alternatively, a single explicit sequence $(\\gamma_q)$ and decreasing $\\psi$ with divergent extra-divergence sum but $m(W(\\psi,\\gamma))<1$ would refute the theorem outright.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The fixed-center inhomogeneous Khintchine theorem that the paper sets out to generalize to moving targets."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic formula for the counting function in the moving-center case, used to prove the fast-divergence proposition and as the baseline for extra divergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the abstraction behind the quasi-independence lemma that absorbs the gcd overlap term under extra divergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the fixed-center quasi-independence estimate and the shifted reduced-denominator sets on which the finitely-centered theorem rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the central limit theorem for the divisor function used in the divisor-count transfer."},{"cited_title":"Cambridge University Press, Cambridge, french edition, 1995","cited_arxiv_id":null,"evidence_quote":"Provides the estimate $\\sum_{n\\le x}1/d(n)\\asymp x/\\sqrt{\\log x}$ used in the divisor-count transfer."},{"cited_title":"Beresnevich, D","cited_arxiv_id":null,"evidence_quote":"Supplies the local full-measure criterion used after Theorem 1 establishes positive measure on every open set."}],"review_version":1}