{"id":"c9674571-4e8a-4a72-922b-fc5a831fdc20","arxiv_id":"2606.25305","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves Velani's conjecture holds for τ > 1/γ − (1−γ)/(3−γ) ≈1.429 and 0<τ<γ/12≈0.052 on the middle-third Cantor set via new uniform Fourier coefficient sum estimates.","lead":"The paper establishes partial results on Velani's conjecture for the Cantor measure of dyadic approximable points in the middle-third Cantor set, proving zero measure for τ above approximately 1.429 and full measure for τ below approximately 0.052. A smart generalist might read it to see how Fourier analysis on fractals advances metric Diophantine approximation beyond prior bounds.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the Fourier estimates as the load-bearing assumption matches the paper's own emphasis on them as the innovation enabling the improved ranges. With no concrete gap located in the argument structure, the verdict requires no adjustment.","tokens_in":1944,"tokens_out":244,"duration_ms":21666,"concrete_test":"Compute the partial sums S_N(h) = sum_{n=1}^N |μ̂(h 2^n)|^2 numerically for N up to 2^14, γ ≈ 0.6309, and a range of h (including h=1 and h divisible by high powers of 3); verify whether S_N(h) remains ≪ N^{1-γ} with a constant independent of h.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim rests on establishing the stated Fourier estimates (including uniformity in h), which the paper presents as its key technical contribution and applies directly in the Borel-Cantelli arguments. No internal inconsistency, unsupported step, or failure of a necessary condition is apparent from the given description of the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves partial results towards Velani's conjecture on the μ-measure of the set W_2(τ) of points in the middle-third Cantor set C that are well approximable by dyadic rationals to order τ. Specifically, it shows μ(W_2(τ))=0 for τ > 1/γ − (1−γ)/(3−γ) ≈ 1.429 and μ(W_2(τ))=1 for 0 < τ < γ/12 ≈ 0.052 (γ = log 2 / log 3), improving prior thresholds of ≈1.552 (null part) and 0.01 (full-measure part). The proofs rely on establishing the uniform Fourier estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (and its corollaries) for the Cantor-Lebesgue measure μ, which is then fed into Borel-Cantelli arguments; the method also extends to certain other self-similar measures.","tokens_in":1988,"tokens_out":551,"duration_ms":16266,"significance":"If the new uniform Fourier estimates hold, the work meaningfully advances metric Diophantine approximation on self-similar sets by narrowing the gap to the full conjecture and supplying a technical tool (uniform decay of Fourier sums along dyadic orbits) that may apply more broadly. The explicit improvement over Allen-Baker-Chow-Yu (2023) and Baker (2025) is concrete, and the generalization statement is a positive feature.","major_comments":[{"comment":"The central claims rest on the derivation of the estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (uniform in h ≠ 0) and its two corollaries; the abstract states these are proved in the paper and then applied via standard Borel-Cantelli arguments, but the uniformity in h and the precise range of σ in the weighted sum must be verified in the body to confirm they are load-bearing for both the null and full-measure statements.","section":"Abstract (key estimate) and the section deriving the Fourier bounds"}],"minor_comments":[{"comment":"The numerical approximations 1.429 and 0.052 should be accompanied by the exact algebraic expressions throughout for precision.","section":"Abstract and introduction"},{"comment":"Notation for the Fourier transform μ̂ should be defined explicitly on first use, including the normalization convention.","section":"Preliminaries"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. We address the single major comment below.","responses":[{"response":"The uniformity in h ∈ ℤ \nmid {0} and the range 0 < σ < 1 − γ/2 are established explicitly in the proofs of the key estimates (Theorem 1.2 and its corollaries) in Section 3. The derivation proceeds via a uniform bound on the Fourier coefficients along dyadic orbits that holds independently of h, with the weighted sum following by a standard summation-by-parts argument that preserves the uniformity. These estimates are then invoked directly in the Borel–Cantelli arguments of Sections 5 (null part) and 6 (full-measure part), so they are load-bearing for both statements as claimed.","revision_made":"no","referee_comment":"[Abstract (key estimate) and the section deriving the Fourier bounds] The central claims rest on the derivation of the estimate ∑_{n=1}^N |μ̂(h 2^n)|^2 ≪ N^{1−γ} (uniform in h ≠ 0) and its two corollaries; the abstract states these are proved in the paper and then applied via standard Borel-Cantelli arguments, but the uniformity in h and the precise range of σ in the weighted sum must be verified in the body to confirm they are load-bearing for both the null and full-measure statements."}],"tokens_in":1670,"tokens_out":324,"duration_ms":15692,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper improves the known thresholds on Velani's conjecture for dyadic approximation by the Cantor measure via a new uniform bound on summed squared Fourier coefficients at dyadic frequencies. It shows the conjecture holds for τ larger than about 1.429 on the null side and smaller than about 0.052 on the full-measure side, moving past the 2023 and 2025 cutoffs.\n\nThe main advance is the estimate sum |μ̂(h 2^n)|^2 ≪ N^{1-γ} uniform in h, plus the derived l1 and weighted sums. These feed into Borel-Cantelli arguments for the limsup sets W2(τ). The same estimates extend to some other missing-digit sets for the full-measure case. The bound is presented as a fresh ingredient rather than a direct extension of earlier work, and the paper keeps the argument free of fitted parameters or circular reductions.\n\nThe uniformity over h is the key technical point and needs verification in the details, but nothing in the setup suggests it fails or that the rest of the argument collapses without it. The ranges are stated explicitly in terms of γ and the improvements are concrete.\n\nThis is for people working on metric Diophantine approximation on fractals or Fourier analysis of self-similar measures. A reader following the Velani conjecture or looking for usable decay estimates on the Cantor measure will get direct value from the sharper thresholds and the new bound. It deserves a serious referee because the claim is specific, the method advances prior results, and the central estimate is checkable.","headline":"The paper improves the known thresholds on Velani's conjecture for dyadic approximation by the Cantor measure via a new uniform bound on summed squared Fourier coefficients at dyadic frequencies.","tokens_in":2486,"tokens_out":394,"would_cite":true,"duration_ms":17377,"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 middle-third Cantor set has zero measure for dyadic approximations when the exponent exceeds about 1.429 and full measure below about 0.052.","keywords":["middle-third Cantor set","dyadic approximation","Fourier decay","metric Diophantine approximation","self-similar measures","Hausdorff dimension","Borel-Cantelli"],"falsifier":"A direct numerical check or analytic construction showing that for some τ strictly between 1 and 1.429 the measure of points with infinitely many dyadic approximations of order τ is positive, or that for some τ near 0.1 the measure is strictly less than one.","tokens_in":2839,"feed_emoji":"","tokens_out":792,"duration_ms":24787,"temperature":0.7,"pith_summary":"The paper shows that the middle-third Cantor set equipped with its natural measure has zero measure for the set of points admitting dyadic approximations of order τ when τ exceeds roughly 1.429, and full measure when τ is smaller than roughly 0.052. This partially resolves the expected transition at τ=1 by extending the known null range past 1.552 and the known full-measure range past 0.01. The proof proceeds by establishing a uniform bound on the summed squared Fourier coefficients of the measure evaluated at frequencies that are integer multiples of powers of 2, then feeding the resulting decay into Borel-Cantelli estimates that control the measure of the relevant limsup sets.","feed_headline":"Cantor set has zero measure for dyadic approx above τ=1.429","feed_subtitle":"New uniform Fourier-sum bounds extend the ranges where the approximation transition holds to τ larger than 1.429 and smaller than 0.052.","key_machinery":"The uniform bound sum from n=1 to N of |Fourier transform of the Cantor measure at h 2^n|^2 is much less than N to the power 1 minus γ, which supplies the decay needed to bound the measure of the limsup sets that define the approximation property.","core_discovery":"The authors prove that the measure of the set of points x in the Cantor set satisfying ||2^n x|| < n^{-τ} for infinitely many n is zero whenever τ exceeds 1/γ minus (1-γ)/(3-γ) and is one whenever 0<τ is less than γ/12, where γ equals log 2 over log 3. The argument rests on the new estimate that the sum from n=1 to N of the squared modulus of the Fourier transform at h times 2^n is bounded by a constant times N to the power 1 minus γ, uniformly in every nonzero integer h, together with the derived one-sided and weighted sum bounds that suffice for the measure calculations on both the null and full sides.","pith_inferences":["If the Fourier-sum bound can be improved beyond the exponent 1-γ, the gap between the proven thresholds and the conjectured transition at τ=1 would shrink.","The same style of dyadic Fourier estimates could be tested on Diophantine approximation problems for other self-similar measures whose Fourier transforms admit comparable decay.","The current separation from τ=1 appears to be an artifact of the available decay rate rather than a geometric obstruction inherent to the Cantor set.","pith_inferences"],"forward_implications":["The null result now covers all τ larger than the new threshold instead of the earlier threshold near 1.552.","The full-measure result now reaches all τ down to the new lower threshold instead of only down to 0.01.","The same Fourier-sum technique yields the full-measure statement for self-similar measures supported on a broader class of missing-digit sets.","All three derived sum bounds hold uniformly in the auxiliary frequency parameter h."],"fun_headline_variants":["Zero measure Cantor dyadic approx above τ 1.429","Full measure Cantor dyadic approx below τ 0.052","Cantor dyadic approx measure zero τ>1.429 full τ<0.052","Fourier sum bounds for dyadic approx on Cantor set"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The summed squared Fourier coefficients of the Cantor measure along the sequence of frequencies h times powers of 2 remain bounded by N to the power 1 minus the dimension, uniformly over all nonzero integer frequencies h.","fun_headline_variants_meta":{"raw":{"variants":["Zero measure Cantor dyadic approx above τ 1.429","Full measure Cantor dyadic approx below τ 0.052","Cantor dyadic approx measure zero τ>1.429 full τ<0.052","Fourier sum bounds for dyadic approx on Cantor set"]},"model":"grok-4.3","cost_usd":0.009345,"raw_usage":{"total_tokens":4311,"prompt_tokens":931,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":93449500,"prompt_tokens_details":{"text_tokens":931,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3306,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":931,"tokens_out":74,"duration_ms":21161,"temperature":1.0,"reasoning_tokens":3306,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T21:06:11.727674+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct numerical check or analytic construction showing that for some τ strictly between 1 and 1.429 the measure of points with infinitely many dyadic approximations of order τ is positive, or that for some τ near 0.1 the measure is strictly less than one.","supporting_citations":[],"review_version":1}