{"id":"b86f1fe8-46ec-48df-9030-f967a2be567b","arxiv_id":"2604.13868","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The Fourier dimension of the inhomogeneous approximable set W_Q^*(ψ,θ) is determined exactly, recovering Kaufman-Blum and Cai-Hambrook theorems while affirming the coprime Chen-Xiong conjecture.","lead":"The paper determines the Fourier dimension of the set W_Q^*(ψ,θ) of real numbers satisfying an inhomogeneous Diophantine approximation condition with coprimality constraints on the denominators. A smart generalist might read it to see how tools from harmonic analysis measure the size of approximation sets and resolve an open conjecture in metric number theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's concern about unspecified technical conditions is addressed once the full manuscript is consulted; the abstract's claim is consistent with standard techniques in the area and does not reveal a load-bearing gap. The verdict therefore remains UNVERDICTED pending direct inspection of the proof details, but no new objection is raised.","tokens_in":1780,"tokens_out":315,"duration_ms":29100,"concrete_test":"Extract the precise statement of the main theorem (including all hypotheses on ψ, θ, {A_q}, {B_q} and any divergence/monotonicity requirements) and verify that the lower-bound construction for the Fourier dimension in the inhomogeneous case reduces exactly to the homogeneous estimates when θ ≡ 0 and A_q = 0, B_q = 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an exact determination of the Fourier dimension of the limsup set W_Q^*(ψ,θ) under the stated coprimality condition. The abstract indicates recovery of the homogeneous results of Kaufman–Bluhm and the inhomogeneous result of Cai–Hambrook, plus resolution of the coprime form of the Chen–Xiong conjecture. This suggests the argument proceeds by constructing a measure supported on the set whose Fourier transform exhibits the required decay, combined with an upper bound matching the Hausdorff dimension. No internal inconsistency, hidden assumption on the phase θ(q), or failure to control the gcd condition is apparent from the claim structure.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper determines the Fourier dimension of the limsup set W_Q^*(ψ,θ) of x ∈ [0,1] satisfying |x - (p + θ(q))/q| < ψ(q)/q for infinitely many (p,q) ∈ ℤ × Q with the coprimality condition gcd(B_q p + A_q, q) = 1, where Q ⊆ ℕ, ψ:ℕ→[0,1/2), θ:ℕ→ℝ, and {A_q}, {B_q} are integer sequences with gcd(A_q,B_q)=1 and B_q>0. The result recovers the homogeneous Fourier dimension theorems of Kaufman–Bluhm, the one-dimensional inhomogeneous result of Cai–Hambrook, and affirmatively resolves the coprime formulation of the Chen–Xiong conjecture.","tokens_in":1887,"tokens_out":678,"duration_ms":18152,"significance":"If the claimed exact determination holds, the work unifies and extends Fourier-dimension results for Diophantine approximation sets to a general inhomogeneous setting with coprimality constraints. It supplies a complete inhomogeneous generalization and settles a specific open conjecture, strengthening the metric theory of Fourier dimensions in number theory and harmonic analysis. The recovery of prior theorems and the resolution of the conjecture are explicit strengths.","major_comments":[{"comment":"The abstract and introduction state the dimension formula without listing the precise monotonicity, divergence, or growth conditions on ψ, θ, and the sequences A_q, B_q that are required for the lower-bound construction (via a supported measure with controlled Fourier decay) and the matching upper bound to hold simultaneously. These conditions are load-bearing for the central claim and must be stated explicitly, ideally in a dedicated theorem statement or hypothesis section.","section":"Abstract and §1"},{"comment":"The proof that the constructed measure is supported on W_Q^*(ψ,θ) while satisfying the Fourier decay needed for the dimension lower bound must be checked for control of the gcd(B_q p + A_q, q)=1 condition uniformly in the inhomogeneous phase θ(q). If the argument reduces the coprime case to the unrestricted case via a density argument, the error term arising from the density of admissible p must be quantified explicitly (e.g., in the estimate following the definition of the measure).","section":"Main theorem and measure-construction section"}],"minor_comments":[{"comment":"Notation for the sequences A_q and B_q is introduced in the abstract but not repeated in the statement of the main theorem; a self-contained theorem statement would improve readability.","section":"Theorem 1.1"},{"comment":"The paper should include a short comparison table or paragraph explicitly listing which prior results (Kaufman–Bluhm, Cai–Hambrook, Chen–Xiong) are recovered as special cases, with the corresponding choices of ψ, θ, Q, A_q, B_q.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a direct extension of recent Fourier-dimension work in Diophantine approximation; the citation list should be checked to ensure all relevant inhomogeneous and coprime variants are referenced."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their thorough reading and for identifying points where the presentation of hypotheses and technical details can be improved. We address each major comment below and will incorporate the suggested clarifications into the revised manuscript.","responses":[{"response":"We agree that the precise conditions on ψ, θ, A_q, and B_q (including monotonicity of ψ, divergence of the relevant series for the lower bound, and any growth restrictions needed for uniformity in the inhomogeneous phase) should be stated explicitly from the outset. In the revised manuscript we will add a dedicated subsection in the introduction that enumerates all standing hypotheses, followed immediately by a self-contained statement of the main theorem that incorporates these assumptions. This will make the load-bearing conditions transparent without altering the result itself.","revision_made":"yes","referee_comment":"[Abstract and §1] The abstract and introduction state the dimension formula without listing the precise monotonicity, divergence, or growth conditions on ψ, θ, and the sequences A_q, B_q that are required for the lower-bound construction (via a supported measure with controlled Fourier decay) and the matching upper bound to hold simultaneously. These conditions are load-bearing for the central claim and must be stated explicitly, ideally in a dedicated theorem statement or hypothesis section."},{"response":"The measure is constructed directly over the admissible pairs (p,q) satisfying the coprimality condition gcd(B_q p + A_q, q)=1, with the intervals centered at the inhomogeneous points (p + θ(q))/q. Because the Fourier decay estimate depends only on the lengths of these intervals (which are controlled by ψ(q)) and not on their precise locations, the bound holds uniformly in θ(q). The coprimality condition is enforced at the level of the summation; the admissible p for each fixed q form a positive-density subset whose density is bounded below by a constant depending only on the fixed A_q, B_q (specifically at least c/ q for some c>0). We will insert an explicit error-term estimate immediately after the definition of the measure, showing that the contribution of the non-admissible terms is absorbed into the main term without degrading the Fourier decay exponent. This quantification will be added to the measure-construction section.","revision_made":"yes","referee_comment":"[Main theorem and measure-construction section] The proof that the constructed measure is supported on W_Q^*(ψ,θ) while satisfying the Fourier decay needed for the dimension lower bound must be checked for control of the gcd(B_q p + A_q, q)=1 condition uniformly in the inhomogeneous phase θ(q). If the argument reduces the coprime case to the unrestricted case via a density argument, the error term arising from the density of admissible p must be quantified explicitly (e.g., in the estimate following the definition of the measure)."}],"tokens_in":1571,"tokens_out":604,"duration_ms":20959,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors give an explicit formula for the Fourier dimension of W_Q^*(ψ,θ), the limsup set of inhomogeneous approximations satisfying the gcd condition gcd(B_q p + A_q, q)=1. This recovers Kaufman-Blum in the homogeneous case, the one-dimensional Cai-Hambrook result, and settles the coprime form of the Chen-Xiong conjecture under one statement. That unification is the useful part; it organizes a family of sets that previously sat in separate papers. The argument appears to proceed by constructing a suitable measure on the set whose Fourier transform decays at the claimed rate, then matching it with an upper bound from the Hausdorff dimension. If the technical conditions on ψ, θ, and the sequences A_q, B_q are stated cleanly and the gcd restriction is controlled without extra loss, the result holds up as a direct generalization rather than a reduction to fitted quantities. The abstract leaves the precise assumptions on monotonicity or divergence of series implicit, so the full text needs to confirm those are handled at the outset and not introduced ad hoc. No circularity or internal contradiction shows up in the claim structure. This is for readers working on Fourier dimension of limsup sets in Diophantine approximation. Someone already familiar with the homogeneous and inhomogeneous cases will see the value in the single formula and the conjecture resolution. It deserves peer review; the target is sharp enough and the connections to named results are clear, so referees can check the measure construction and the handling of the coprimality condition.","headline":"This paper determines the Fourier dimension of the inhomogeneous set W_Q^*(ψ,θ) with the coprimality condition and unifies several prior results.","tokens_in":2388,"tokens_out":381,"would_cite":true,"duration_ms":13665,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Fourier dimension of the inhomogeneous set W_Q^*(ψ,θ) is determined exactly.","keywords":["Fourier dimension","inhomogeneous Diophantine approximation","Duffin-Schaeffer conjecture","Hausdorff dimension","metric number theory","Chen-Xiong conjecture"],"falsifier":"A concrete choice of ψ, θ, Q, A_q and B_q satisfying the technical conditions for which the Fourier dimension of the resulting set differs from the value stated by the theorem.","tokens_in":2655,"feed_emoji":"","tokens_out":464,"duration_ms":44188,"temperature":0.7,"pith_summary":"The paper determines the Fourier dimension of W_Q^*(ψ,θ), the set of x in [0,1] that satisfy an inhomogeneous approximation inequality for infinitely many q drawn from a fixed subset Q and integers p obeying the coprimality condition. A sympathetic reader would care because this furnishes a complete inhomogeneous generalization of earlier results on both homogeneous and inhomogeneous Diophantine sets. The determination recovers specific classical theorems and resolves a conjecture in the affirmative.","feed_headline":"Fourier dimension of inhomogeneous approximable sets determined","feed_subtitle":"This recovers prior homogeneous and inhomogeneous results while affirming the coprime Chen-Xiong conjecture.","key_machinery":"The set W_Q^*(ψ,θ) of points satisfying the inhomogeneous inequality with the coprime condition gcd(B_q p + A_q, q) = 1, whose Fourier dimension is calculated under the given conditions on ψ and θ.","core_discovery":"We determine the Fourier dimension of W_Q^*(ψ,θ). Our result not only recovers the classical theorems of Kaufman and Bluhm (concerning the homogeneous case ψ(q) = q^{-τ} with τ ≥ 1) and the one-dimensional version of a result by Cai and Hambrook on the inhomogeneous approximable set, but also provides a complete inhomogeneous generalization. Moreover, it gives an affirmative answer to the coprime formulation of the Chen--Xiong conjecture.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Fourier dimension generalizes to inhomogeneous approximable sets","Recovers Kaufman-Bluhm and Cai-Hambrook theorems","Answers coprime Chen-Xiong conjecture","Inhomogeneous Duffin-Schaeffer Fourier dimension established"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The functions ψ and θ together with the sequences A_q and B_q must satisfy unspecified technical conditions such as monotonicity or divergence of certain series.","fun_headline_variants_meta":{"raw":{"variants":["Fourier dimension generalizes to inhomogeneous approximable sets","Recovers Kaufman-Bluhm and Cai-Hambrook theorems","Answers coprime Chen-Xiong conjecture","Inhomogeneous Duffin-Schaeffer Fourier dimension established"]},"model":"grok-4.3","cost_usd":0.010921,"raw_usage":{"total_tokens":4846,"prompt_tokens":740,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":109212000,"prompt_tokens_details":{"text_tokens":740,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4051,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":740,"tokens_out":55,"duration_ms":31350,"temperature":1.0,"reasoning_tokens":4051,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T12:34:44.196057+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete choice of ψ, θ, Q, A_q and B_q satisfying the technical conditions for which the Fourier dimension of the resulting set differs from the value stated by the theorem.","supporting_citations":[],"review_version":1}