{"id":"0a563abc-0535-4649-b948-702959bdb66e","arxiv_id":"2506.17031","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new theorem shows that a small-scale additive energy estimate plus a weighted large-gap sum implies metric Poissonian pair correlation, yielding the property for quantitatively convex and real polynomial sequences.","lead":"This paper proves new conditions under which real sequences have metric Poissonian pair correlation, meaning their scaled fractional parts look randomly distributed at small scales. It improves a 2021 result, lowers an energy threshold, and answers open questions for convex and polynomial sequences.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.3 applies Proposition 4.6 with J=N0^2/L^2 although Proposition 4.5 only gives L≤4N0; J can be <1, and Proposition 4.6 does not cover this case.","rationale":"The paper's central claim is Theorem 2.1, whose proof rests on the lattice-point-counting bound Theorem 4.1. The apparent gap in Section 4.3 is load-bearing because Proposition 4.6 and Lemma 4.2 are applied in a regime not covered by their hypotheses. The Reader's weakest assumption was condition (2.3), and the Reader's localized concern was the arithmetic slip in Proposition 9.1; neither identifies this dyadic-length issue. In good faith, I do not assert the theorem is false: the gap may be repairable with a modified argument and constants, as sketched in the concrete test. But the manuscript as written does not supply that argument, so the central claim is not fully verified. I therefore recommend UNVERDICTED rather than a flat REJECT, and I keep the analysis free of any judgment about the authors beyond the technical discrepancy.","tokens_in":26356,"tokens_out":62941,"duration_ms":612225,"concrete_test":"Re-derive Section 4.3 with L=2N0, a case allowed by Proposition 4.5. For X_k={x}, α(x)=1, K/x=16, N large, and J=N0^2/L^2=1/4, evaluate both sides of the inequality used in the proof: the left side is about 32N while the right side after the factor 1/J is about 4N, so Proposition 4.6 cannot justify the step. Then check whether replacing that step by the direct dyadic bound eS(X_k,θ1L^2,LK') ≪ N^{o(1)} S(X,α,256N0^2,8N0^2K/N), with N0≈N^{1/2}/C chosen so that 256N0^2≤N/2, still yields the final N^{4−δ} bound in Theorem 2.1; if not, the proof of the main theorem has a genuine gap.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"At Section 4.3, after Proposition 4.5 the dyadic length L is only known to satisfy L≤4N0. The proof then invokes Proposition 4.6 with J=N0^2/L^2 and Lemma 4.2 with parameter LN/(N0K). Both require L≤N0 up to constants. When N0<L≤4N0, J lies in [1/16,1), outside the hypothesis \"J≫1\" of Proposition 4.6; the proof of that proposition needs a prime in [4J,8J], which is impossible for J<1/4. The inequality itself is not valid for J<1 with the stated factor: taking X_k={x}, α(x)=1, K/x=1/J, one gets eS(X_k,N,K)≈2N/J while (1/J)eS(X_k,JN,JK)≈2N, so the ratio grows like 1/J as J→0. The L>N0 case is not treated separately, so Theorem 4.1, and hence Theorem 2.1 via Section 8.1, is not established as written. The Reader's arithmetic slip in Proposition 9.1 is real but less central: the corrected exponent 12427/5000 is still below 5/2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new sufficient condition for a real sequence (x_n) to be metric Poissonian. Theorem 2.1 requires a weakened gap condition x_{n+1}-x_n >= c n^{eta-1} with eta in (2/3,1], an approximate-energy bound E*_{N,1/N} << N^{3-kappa-delta}, and a weighted large-gap sum bound. The proof reduces the metric Poissonian problem, via a theorem of Rudnick and Technau, to a lattice point counting estimate S(X,alpha,M,K), which is then attacked through a sequence of recursive inequalities (Propositions 4.3-4.7). Applications are given to quantitatively convex sequences and to polynomial sequences, using energy estimates proved with results of Bloom and of Bourgain-Demeter-Guth and Wooley.","tokens_in":26619,"tokens_out":17670,"duration_ms":161385,"significance":"If the proof is completed, the results constitute a substantial advance over Aistleitner-El-Baz-Munsch: they allow gaps that decay polynomially rather than requiring a uniform lower bound, and they replace the coarse energy E*_N by the finer E*_{N,1/N}. The paper also answers an open problem of Aistleitner-El-Baz-Munsch under the additional weight condition (2.3), and it gives the first metric Poissonian results for general convex and polynomial sequences. The recursive lattice-pointing-counting framework is a genuinely different technical route from earlier work. However, the current version contains a load-bearing gap in the proof of Theorem 4.1 and an arithmetic slip in Proposition 9.1; these need to be repaired before the main claims are established.","major_comments":[{"comment":"The proof of Theorem 4.1 applies Proposition 4.6 with J = N_0^2/L^2 after Proposition 4.5 has only produced L <= 4N_0. When N_0 < L <= 4N_0, we have J in [1/16,1), so the hypothesis J >> 1 of Proposition 4.6 is not satisfied; its proof requires a prime q in [4J,8J], which does not exist for J < 1/4, and the claimed inequality eS(X_k,alpha,N,K) << J^{-1} eS(X_k,alpha,theta J N, J K) is in fact false for J < 1 in general, e.g. with X_k={x}, alpha(x)=1, K/x=1/J one gets LHS of order N/J and RHS of order N. The subsequent applications of Lemma 4.2 and Proposition 4.4 with parameters LN/(N_0K) and N_0/L also require L <= N_0. Since the case N_0 < L <= 4N_0 is not treated, Theorem 4.1 is not established as written, and therefore the application of Theorem 4.1 in Section 8.1 to prove (8.10) is unsupported. The authors should either prove a version of Proposition 4.6 valid for J in [c,1) for some c>0, ensure that L can be chosen with L <= N_0, or handle the large-L case by a separate argument.","section":"Section 4.3, paragraph applying Proposition 4.6"},{"comment":"The displayed chain after (9.7) loses a power of K. From Lemma 9.3 and Lemma 9.4 one obtains E(X)^{1/4} << N^{o(1)} K^{77/200} N^{123/200}; raising to the fourth power and using (9.6) gives E*_N << N^{1/100} K^{77/50} N^{123/50+o(1)} = N^{12427/5000+o(1)} because K = N^{1/100}, not N^{247/100+o(1)} as stated. The corrected exponent 12427/5000 is still smaller than 5/2, so Corollary 2.4 remains valid, but the statement of Proposition 9.1 and the surrounding calculation should be corrected.","section":"Section 9.1, after Eq. (9.7)"}],"minor_comments":[{"comment":"The displayed definition alpha(k) = 4/(k(k+1)) - 2 appears to be a typo: the Hölder interpolation immediately before it requires alpha(k) = 4/(k(k+1)-2). With the printed definition the subsequent exponent computation does not go through.","section":"Section 9.2, Eq. (9.9)"},{"comment":"The displayed bound for S(X_+,alpha,N^{1+epsilon},N^epsilon) omits the factors N^{O(epsilon)} coming from K^{1+o(1)}, (MK)^{1/2+o(1)} and (MK)^{1+o(1)} in Theorem 4.1. These can be absorbed by choosing epsilon sufficiently small in Proposition 8.1, but the display should either retain them or explicitly state this convention.","section":"Section 8.1, display after applying Theorem 4.1"},{"comment":"There are several minor typographical issues, e.g. 'there exits' in Theorem 3.1 and 'Zaharescru' in the introduction; these should be corrected in a final revision.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The gap in Section 4.3 is the main obstacle; Theorem 2.1 and the applications depend on Theorem 4.1. I believe the gap is repairable, but until it is fixed the main theorem is unproved. The arithmetic slip in Proposition 9.1 is real but, as noted, the corrected exponent still supports the relevant corollary. I do not see any circularity concern: the external results cited are standard and do not encode the target theorem. If the Section 4.3 issue cannot be repaired, the paper would need to be rejected; otherwise, after the stated corrections, it would be a strong contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you my take on the Kerr–Wang paper. The headline: this is a real step forward for metric pair correlation — the energy threshold improves from 183/76 to 5/2, the spacing condition relaxes to allow n^{η−1} gaps with η>2/3, and the applications to quantitatively convex and real polynomial sequences answer open questions in [1]. The new Heath-Brown-style recursive inequalities for the lattice sums S(X,α,M,K) are genuinely novel and likely to be reused.\n\nBut there is a serious gap in the proof of the central lattice-counting theorem. In Section 4.3, the proof invokes Proposition 4.6 with J = N0^2/L^2, yet Proposition 4.5 only guarantees L ≤ 4N0. If L > N0, then J < 1, and the hypothesis J ≫ 1 of Proposition 4.6 is not met. This is not pedantic: the proof of Proposition 4.6 needs a prime q in [4J,8J], and for J < 1/4 there is none. The L > N0 case is not treated separately. Since the preliminary bound (4.6) feeds directly into the final bound for Theorem 4.1, and Theorem 2.1 relies on Theorem 4.1 via Section 8.1, the main results are not established as written. This looks fixable — one could choose N0 relative to L or add a direct argument for L > N0 — but it is a load-bearing technical issue, not a typo.\n\nThere is also a smaller arithmetic slip in Proposition 9.1: after equation (9.7), the exponent should be 12427/5000 = 2.4854, not 247/100 = 2.47. The corrected bound is still below 5/2, so Corollary 2.4 survives, but the stated exponent is wrong.\n\nOn the positive side, the energy estimates for polynomials via a Hua-type inequality and the use of Bloom's convex-energy bound are clean. The citation pattern is proper: the external tools (Rudnick–Technau, Bloom, Bourgain–Demeter–Guth, Wooley) are all used appropriately. The weighted large-gap condition (2.3) is admittedly technical and the authors themselves flag it as probably weakenable; that's a limitation rather than a defect.\n\nWho is this for? Anyone working on metric pair correlation and lattice point problems. The recursive inequalities have independent value. It deserves a serious referee, but the referee should insist on a corrected Section 4.3.\n\nRecommendation: send to peer review. With the J issue repaired, it would be a strong addition to the literature.","headline":"Good ideas and genuinely new results, but a real gap in the proof of Theorem 4.1 (the J = N0^2/L^2 issue) means the main theorems are not yet established as written.","tokens_in":27146,"tokens_out":8714,"would_cite":false,"duration_ms":74735,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K06","11K31","11B30","11L07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A real sequence is metric Poissonian when its approximate additive energy and its large-gap sums satisfy two power bounds; quantitatively convex and polynomial sequences meet them.","keywords":["metric Poissonian pair correlation","approximate additive energy","real sequences","lattice point counting","Dirichlet polynomial mean values","convex sequences","polynomial sequences","pair correlation"],"falsifier":"One concrete test is to compute, for the sequence x_n = $n^{{3/4}}$, the quantities E*_{N,1/N} and the large-gap sum in (2.3) for growing N, and to compare the pair correlation of (α $n^{{3/4}}$) against the Poisson limit 2s; if the pair correlation visibly fails while both numerical bounds hold, Theorem 2.1 would be false. A more direct check for any candidate sequence is the lattice-counting condition from the reduction theorem: if the number of solutions to |m1(xn1−xn2)−m2(xn3−xn4)| < N^ε exceeds $N^{{4−δ}}$ for every δ > 0, the sequence cannot be metric Poissonian.","tokens_in":26137,"feed_emoji":"🔢","tokens_out":6690,"duration_ms":65581,"temperature":0.7,"pith_summary":"This paper establishes new sufficient conditions for a real sequence to be metric Poissonian: scaled copies of the sequence have pair correlations matching a Poisson process for almost every scaling factor. The main theorem requires three quantitative hypotheses: gaps cannot shrink too fast, an approximate additive energy E*_{N,1/N} must fall a power below $N^{3}$, and a weighted count of large gaps must grow mildly. If these hold, the fractional parts are pseudo-random at the pair-correlation scale. As applications, quantitatively convex sequences and all real polynomials of degree at least two are shown to be metric Poissonian, answering open questions from earlier work.","feed_headline":"Three conditions force Poisson pair correlations","feed_subtitle":"New energy and gap bounds make quantitatively convex and polynomial sequences Poissonian for almost every scale.","key_machinery":"The argument reduces metric Poissonian pair correlation to counting lattice points in regions defined by inequalities |m1x1−m2x2| ≤ K, where x1 and x2 are differences of the original sequence. The central objects are the approximate additive energy E*_{N,γ}, counting quadruples with |xn1−xn2+xn3−xn4| < γ, and the weighted large-gap sum appearing in (2.3). A set of recursive inequalities, derived from geometry of numbers and mean values of Dirichlet polynomials, lets the proof inflate and shrink the parameter K while keeping the relevant counting problem under control; the final bound is expressed directly in the two quantities named in the theorem.","core_discovery":"The central claim is Theorem 2.1: for a positive real sequence with gaps xn+1−xn ≥ c $n^{{η−1}}$ for some η in (2/3,1], the approximate energy bound E*_{N,1/N} ≪ $N^{{3−κ−δ}}$, and the weighted large-gap estimate Σ_{1≤n1<n2≤N, xn2−xn1≥1} (xn2−xn1)^{−1/2} ≪ $N^{{1+κ/2}}$, together imply that the sequence is metric Poissonian. This improves the previous bounded-gap energy threshold from $N^{{183/76−δ}}$ to $N^{{5/2−δ}}$, and under a mild extra assumption it answers positively the open question whether a γ≈1/N energy condition suffices. The proof goes through a lattice-point counting reduction and a system of recursive inequalities for weighted sums over differences, ultimately controlling the relevant counting problem in terms of the approximate energy and the large-gap sum.","pith_inferences":["Beyond the paper's claims, the condition η > 2/3 is described as technical and likely not sharp; a sharper recursive inequality could plausibly lower it, perhaps toward the point where the energy and large-gap bounds naturally balance.","Condition (2.3) is probably not necessary for the metric Poissonian property; the proof uses it only to dominate one error term, so sequences with many large gaps but still Poissonian correlations are plausible candidates for a refined theorem.","The same energy-estimate machinery could be tested on divisor-sum sequences of the form σ_β(n), which the paper mentions but does not resolve; a first step would be computing the analogue of E*_{N,1/N} for such sequences.","The lattice-point bound Theorem 4.1 is a standalone estimate for weighted difference sets and may apply to other problems where approximate additive energy and large-gap structure are the only relevant statistics."],"forward_implications":["Any sequence satisfying conditions (2.1), (2.2), and (2.3) has pair correlation converging to the Poisson limit 2s for almost every α.","For sequences with a uniform positive gap, the energy bound E*_N ≪ N^{5/2−δ} alone is sufficient, improving the earlier 183/76 threshold.","Quantitatively convex sequences, whose gaps increase by at least c n^{−1/10^4}, are metric Poissonian.","Every real polynomial of degree at least two gives a metric Poissonian sequence.","The energy estimates for convex and polynomial sequences are stated independently and can serve as tools outside the pair-correlation problem."],"supporting_citations":[{"why":"Supplies the lattice-point counting reduction that turns metric Poissonian pair correlation into estimates for sums of the form S(X,α,M,K).","marker":"[24]"},{"why":"Defines the approximate energy E*_N, gives the previous 183/76 exponent, and states the open problem answered under an additional assumption.","marker":"[1]"},{"why":"Establishes the additive-energy criterion for integer sequences that motivates the real-sequence analogue pursued here.","marker":"[4]"},{"why":"Provides the large-values estimate for Dirichlet polynomials used to prove the increasing property for the lattice sums.","marker":"[15]"},{"why":"Supplies the recursive inequalities for lattice sums that inspire the decreasing and multiplicative properties used in Theorem 4.1.","marker":"[16]"},{"why":"Gives the mean-value identity connecting lattice sums to integrals of Dirichlet polynomials, used in Lemma 7.1.","marker":"[30]"},{"why":"Supplies the additive-energy bound for convex sets used in the proof of Corollary 2.4.","marker":"[7]"},{"why":"Provides the resolution of the main conjecture in Vinogradov's mean value theorem used in the energy estimate for polynomials.","marker":"[11]"},{"why":"Supplies the Vinogradov mean value bounds used in the Hua-type inequality for polynomial exponential sums.","marker":"[31]"}],"fun_headline_variants":["New energy bound yields Poissonian pair correlations","Gap and energy conditions settle open problem","Improved energy threshold for metric Poissonian property","Weak energy condition ensures Poissonian behavior","Energy estimate resolves Poissonian open question"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weighted large-gap bound (2.3) is the most delicate premise: it enters at one step of the final estimate, the paper notes it may be weakened, and a sequence violating it falls outside the theorem even if it is metric Poissonian.","fun_headline_variants_meta":{"raw":{"variants":["New energy bound yields Poissonian pair correlations","Gap and energy conditions settle open problem","Improved energy threshold for metric Poissonian property","Weak energy condition ensures Poissonian behavior","Energy estimate resolves Poissonian open question"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1470,"prompt_tokens":779,"completion_tokens":691,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":395,"completion_tokens_details":{"reasoning_tokens":627}},"tokens_in":395,"tokens_out":691,"duration_ms":7291,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:13:51.487140+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete test is to compute, for the sequence x_n = $n^{{3/4}}$, the quantities E*_{N,1/N} and the large-gap sum in (2.3) for growing N, and to compare the pair correlation of (α $n^{{3/4}}$) against the Poisson limit 2s; if the pair correlation visibly fails while both numerical bounds hold, Theorem 2.1 would be false. A more direct check for any candidate sequence is the lattice-counting condition from the reduction theorem: if the number of solutions to |m1(xn1−xn2)−m2(xn3−xn4)| < N^ε exceeds $N^{{4−δ}}$ for every δ > 0, the sequence cannot be metric Poissonian.","supporting_citations":[{"cited_title":"Rudnick and N","cited_arxiv_id":null,"evidence_quote":"Supplies the lattice-point counting reduction that turns metric Poissonian pair correlation into estimates for sums of the form S(X,α,M,K)."},{"cited_title":"Aistleitner, D","cited_arxiv_id":null,"evidence_quote":"Defines the approximate energy E*_N, gives the previous 183/76 exponent, and states the open problem answered under an additional assumption."},{"cited_title":"Aistleitner, G","cited_arxiv_id":null,"evidence_quote":"Establishes the additive-energy criterion for integer sequences that motivates the real-sequence analogue pursued here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the large-values estimate for Dirichlet polynomials used to prove the increasing property for the lattice sums."},{"cited_title":"184 (2018), no","cited_arxiv_id":null,"evidence_quote":"Supplies the recursive inequalities for lattice sums that inspire the decreasing and multiplicative properties used in Theorem 4.1."},{"cited_title":"Watt, Exponential sums and the Riemann zeta-function","cited_arxiv_id":null,"evidence_quote":"Gives the mean-value identity connecting lattice sums to integrals of Dirichlet polynomials, used in Lemma 7.1."},{"cited_title":"Bourgain, C","cited_arxiv_id":null,"evidence_quote":"Provides the resolution of the main conjecture in Vinogradov's mean value theorem used in the energy estimate for polynomials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Vinogradov mean value bounds used in the Hua-type inequality for polynomial exponential sums."}],"review_version":2}