{"id":"55f14070-dcb9-4da1-a069-d68f14b93389","arxiv_id":"2606.03781","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves Poissonian pair correlation of {α n^θ} for Lebesgue-almost all θ in (0,3/5) ∪ (3,∞) via interval splitting and zeta-moment counting estimates.","lead":"The paper proves that the fractional parts {α n^θ} show Poissonian pair correlation for almost all θ in (0, 3/5) union (3, ∞). This widens the range of exponents where the sequence behaves like a random Poisson process compared to earlier work requiring θ > 7.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Control of exceptional θ-intervals via zeta moments and exponent pairs is the least secure step for extending below θ=7","rationale":"The reader's weakest_assumption correctly isolates the only place where the improvement from 7 to 3/5 could break; without the full text the precise constants in the zeta-moment and exponent-pair steps cannot be checked, so the verdict remains UNVERDICTED pending that verification.","tokens_in":1645,"tokens_out":359,"duration_ms":14748,"concrete_test":"Fix the splitting length δ and the moment order used in the paper; recompute the measure of E by inserting the explicit exponent-pair bound (A,B) cited in the counting lemma and check whether |E| remains o(1) when θ is taken in (0.5, 0.6); if the resulting measure exceeds the target o(1) threshold the Poissonian conclusion fails for a positive-density set of θ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument splits the θ-integral over short intervals I_k of length δ and claims that on all but a negligible-measure subset E the contribution to the variance is O(1) using only |φ'(n,θ)| ≫ 1 (first derivative of the phase α n^θ). The set E is controlled by reducing to Diophantine counting problems whose bounds come from zeta-moment estimates and exponent pairs. For the claimed range down to 3/5 this reduction must produce |E| = o(1) uniformly in the splitting parameters; any loss in the exponent-pair constants or in the moment range would make the exceptional contribution non-negligible precisely when θ drops below the previous threshold of 7.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for fixed α > 0 the sequence {α n^θ} has Poissonian pair correlation for Lebesgue-almost every θ ∈ (0, 3/5) ∪ (3, ∞). The argument splits the θ-integral appearing in the variance into short intervals of length δ, shows that on all but a negligible-measure exceptional set E the contribution is O(1) using only the first derivative of the phase, and controls |E| by reducing to Diophantine counting problems whose bounds are obtained from moments of the Riemann zeta function together with exponent-pair estimates. This improves the earlier threshold θ > 7 obtained by Technau–Yesha via a repulsion principle based on the fourth derivative.","tokens_in":1807,"tokens_out":501,"duration_ms":21115,"significance":"If the counting estimates hold with the required uniformity, the result substantially enlarges the set of θ for which Poissonian pair correlation is known to hold almost everywhere. The technique of handling most intervals with the first derivative and controlling the exceptional set via zeta moments is technically novel and may apply to other metric problems in uniform distribution. The paper supplies a self-contained proof once the cited zeta-moment and exponent-pair bounds are granted.","major_comments":[{"comment":"The control of the exceptional set E (final paragraph of the abstract and the corresponding reduction in the proof of the main theorem): the zeta-moment and exponent-pair estimates must be shown to produce |E| = o(1) uniformly in the splitting parameter δ down to θ = 3/5. Any loss in the admissible range of the moment or in the exponent-pair constant would render the contribution of E non-negligible precisely in the new range below the previous threshold of 7, which is the load-bearing step for the claimed improvement.","section":"proof of main theorem (reduction to counting estimates)"}],"minor_comments":[{"comment":"Notation for the phase function φ(n, θ) and the precise definition of the short intervals I_k should be stated explicitly at the beginning of the splitting argument.","section":"§3 (splitting argument)"},{"comment":"The dependence of the implied constants on α should be tracked through the estimates, even if α is fixed.","section":"counting estimates"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for isolating the uniformity of the exceptional-set bound as the load-bearing step. We address the comment directly below and will revise the manuscript accordingly.","responses":[{"response":"We agree that explicit verification of this uniformity is necessary to justify the improvement below θ = 7. The reduction to the counting estimates (Proposition 3.2) and their proof via zeta-moment bounds and exponent pairs (Section 4) are written so that the resulting |E| is o(1) uniformly in δ for all θ ≥ 3/5; the exponents arising from the cited moment and pair estimates are strictly negative in this range and absorb the δ-dependence. Nevertheless, the dependence on θ and δ is not written out in a single displayed calculation. We will therefore add a short paragraph immediately after the statement of the main theorem that extracts the admissible range from the moment and pair constants and confirms |E| = o(1) uniformly down to θ = 3/5. This will be included in the revised version.","revision_made":"yes","referee_comment":"[proof of main theorem (reduction to counting estimates)] The control of the exceptional set E (final paragraph of the abstract and the corresponding reduction in the proof of the main theorem): the zeta-moment and exponent-pair estimates must be shown to produce |E| = o(1) uniformly in the splitting parameter δ down to θ = 3/5. Any loss in the admissible range of the moment or in the exponent-pair constant would render the contribution of E non-negligible precisely in the new range below the previous threshold of 7, which is the load-bearing step for the claimed improvement."}],"tokens_in":1341,"tokens_out":372,"duration_ms":25480,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main advance is a larger set of θ where the sequence has Poissonian pair correlation for almost every θ. They replace the fourth-derivative repulsion from Technau-Yesha with a split of the θ-integral into short intervals, then bound the variance on most intervals using only the first derivative of the phase. The exceptional intervals reduce to counting problems solved via zeta moments and exponent pairs.\n\nThis is a direct improvement on the threshold. The method is straightforward once the splitting is set up, and the reliance on existing zeta-moment and exponent-pair results keeps the argument self-contained.\n\nThe soft spot is the size of the exceptional set E. The argument needs |E| small enough uniformly down to θ=3/5; any slack in the moment bounds or exponent-pair constants could make the contribution from E non-negligible precisely in the new range below 7. The abstract claims the estimates close this gap, but the constants matter and would need checking in a full read.\n\nThis is for analytic number theorists working on uniform distribution and pair correlations. A reader familiar with zeta moments will see the value quickly. It is worth sending to a referee who can verify the exceptional-set control.","headline":"This extends the Poissonian pair correlation range for αn^θ to almost all θ in (0,3/5)∪(3,∞) by splitting θ-integrals and controlling most pieces with first-derivative bounds plus zeta-moment estimates.","tokens_in":2268,"tokens_out":338,"would_cite":false,"duration_ms":15052,"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 sequence {α n^θ} has Poissonian pair correlation for Lebesgue-almost all θ in (0, 3/5) ∪ (3, ∞).","keywords":["pair correlation","Poissonian distribution","fractional parts","Diophantine approximation","Riemann zeta function","exponent pairs","number theory"],"falsifier":"A positive-measure set of θ inside (0, 3/5) for which the pair-correlation statistic fails to approach the Poisson limit.","tokens_in":2540,"feed_emoji":"","tokens_out":689,"duration_ms":27962,"temperature":0.7,"pith_summary":"For fixed α > 0 the fractional parts α n^θ mod 1 are shown to have Poissonian pair correlation for Lebesgue-almost every θ in the interval (0, 3/5) or for every θ > 3. Poissonian pair correlation means the number of pairs of points lying a given distance apart matches the count expected from a Poisson point process on the circle. The result widens the range of θ where this random-like behavior is known to hold, lowering the previous threshold of 7. The argument splits the integral that computes the variance of the pair-correlation statistic into short intervals in θ and shows that all but a negligible proportion of those intervals can be bounded using only the first derivative of the phase function.","feed_headline":"α n^θ shows Poisson pair correlation for almost all θ","feed_subtitle":"Holds Lebesgue-almost everywhere in (0,3/5) and above 3, improving prior threshold of 7","key_machinery":"Splitting the θ-integration in the variance into many short intervals and bounding most integrals with the first derivative of the phase, reducing the remainder to counting estimates proved via moments of the Riemann zeta function and exponent pairs.","core_discovery":"For fixed α>0, we show that the sequence {α n^θ} has Poissonian pair correlation for Lebesgue-almost all θ ∈ (0,3/5)∪(3,∞).","pith_inferences":["The short-interval splitting technique may extend to pair correlations of other sequences whose phase has a power-law dependence on n.","Sharper bounds on zeta moments or exponent pairs could enlarge the interval (0, 3/5) or lower the threshold 3.","The method replaces the need for a fourth-derivative repulsion principle with first-derivative estimates on most of the measure."],"forward_implications":["The pair-correlation function of {α n^θ} matches the Poisson prediction for almost all θ in the stated ranges.","The variance integral over θ is o(1) once the exceptional intervals are removed.","Counting estimates derived from zeta moments and exponent pairs are sufficient to control the exceptional set.","The same conclusion holds for any fixed α > 0."],"fun_headline_variants":["Poisson pair correlation for α n^θ at almost all θ in (0,3/5) and above 3","α n^θ has Poissonian pair correlation for Lebesgue almost all θ in new range","Almost all θ in (0,3/5) and (3,∞) give Poisson pair correlation to α n^θ","Lebesgue almost all θ in (0,3/5) and (3,∞) give Poisson pair correlation for α n^θ"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That after splitting the θ-integral into short intervals, the contribution from all but a negligible set of intervals can be bounded using only the first derivative of the phase, with the exceptional set controlled by the cited zeta-moment and exponent-pair estimates.","fun_headline_variants_meta":{"raw":{"variants":["Poisson pair correlation for α n^θ at almost all θ in (0,3/5) and above 3","α n^θ has Poissonian pair correlation for Lebesgue almost all θ in new range","Almost all θ in (0,3/5) and (3,∞) give Poisson pair correlation to α n^θ","Lebesgue almost all θ in (0,3/5) and (3,∞) give Poisson pair correlation for α n^θ"]},"model":"grok-4.3","cost_usd":0.011097,"raw_usage":{"total_tokens":4828,"prompt_tokens":563,"num_sources_used":0,"completion_tokens":118,"cost_in_usd_ticks":110974500,"prompt_tokens_details":{"text_tokens":563,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4147,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":563,"tokens_out":118,"duration_ms":22460,"temperature":1.0,"reasoning_tokens":4147,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T08:30:09.866586+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A positive-measure set of θ inside (0, 3/5) for which the pair-correlation statistic fails to approach the Poisson limit.","supporting_citations":[],"review_version":1}