{"id":"7d98ddb0-3f1a-4a29-9b36-5067f19d8c0a","arxiv_id":"2607.10773","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Weak inhomogeneous pair correlation neither implies equidistribution nor compares across distinct offsets γ, unlike the homogeneous case.","lead":"This math paper constructs sequences with weak inhomogeneous Poisson pair correlation that are not equidistributed, and claims that distinct inhomogeneity parameters yield mutually independent weak pair-correlation notions. The first two theorems appear sound; the third has a construction error and a missing concentration argument.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3 Step 2 applies Lemma 3.1 at a scale that shrinks to 0, while the lemma is proved only for fixed s>0; without a uniform-in-s concentration estimate the claimed γ1-PPC limit is unsupported.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: Lemma 3.1 is applied in Theorem 1.3 Step 2 at a variable scale a_r→0, which is outside the lemma's stated fixed-s hypothesis. This gap directly threatens the central claim that Y has weak γ1-PPC. The second premise flagged by the reader, the false invariant (4.1), is also present, but it is less decisive because the proof's counting can be reinterpreted using the true multiplicity property (each x_k and x_k+γ2 appears exactly r times in the first M_r terms). The variable-scale issue, by contrast, would require a new concentration argument even after repairing the construction. The proposed concrete test—checking whether the variance estimate is uniform in s and whether the a_r→0 limit follows—would settle whether the proof can be repaired or whether the theorem lacks support. Since the reader already rejected for essentially this reason, the verdict remains unchanged.","tokens_in":11961,"tokens_out":18073,"duration_ms":165099,"concrete_test":"Re-derive the variance estimate in Lemma 3.1 tracking the s-dependence: prove whether Var R_N(γ,s,δ) ≤ C s/N^{2−δ} holds uniformly for s∈(0,1]. Then use Chebyshev along K_r with a_r = s(K_r/N_r)^δ to check whether R_{K_r}(γ,a_r,δ) = 2a_r + o(a_r) almost surely. If the uniform bound holds, Step 2 is justified; if the bound degenerates as s→0, the γ1-limit in Theorem 1.3 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.3, Step 2, needs the pair correlation of the auxiliary i.i.d. sequence (x_k)_{1≤k≤K_r} at radius s/N_r^δ. Writing this as Lemma 3.1 applied to (x_k) requires the effective scale a_r = s(K_r/N_r)^δ ≈ s(2r)^{-δ}, which tends to 0 as r→∞. Lemma 3.1 is stated and proved only for fixed s>0: the Chebyshev/Borel–Cantelli argument is run for each fixed s, and the cited standard approximation [4, p.475] passes from N_m=m^2 to full N for that same fixed s. It does not control the simultaneous limit r→∞ with a_r→0. The displayed equality 'Σ ... = (K_r(K_r−1)/(N_r(N_r−1))) N_r^{2−δ}(2s+o(1))' therefore assumes an unproved uniformity. If the remainder is only o(1) for each fixed a but not o(a_r), the contribution to R_Y is not 2s+o(1). A second defect exists: the prefix invariant (4.1) is false (e.g., for N=2 the induction gives x1, x1+γ2, x1, x1+γ2 rather than x1^2,(x1+γ2)^2), though the multiplicity statement 'r copies of each x_k and x_k+γ2' is true and might allow a repair.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weak inhomogeneous Poissonian pair correlation (weak γ-PPC) for γ∈(0,1/2] and 0<δ<1. Theorem 1.1 constructs, for every γ and δ, a sequence satisfying weak γ-PPC but not γ-PPC, by alternating an i.i.d. uniform sequence with its γ-shift. Theorem 1.2 constructs a non-equidistributed sequence with weak γ-PPC, using a probabilistic lemma (Lemma 3.1) together with an explicit choice of density g with prescribed autocorrelation. Theorem 1.3 claims that for distinct γ1,γ2∈(0,1/2], weak γ1-PPC does not imply weak γ2-PPC; the proof uses a block construction based on an i.i.d. sequence and aims to show that the γ1 pair-correlation tends to 2s while the γ2 pair-correlation is bounded away from 2s along a subsequence. Theorems 1.1 and 1.2 appear essentially sound, but the proof of Theorem 1.3 contains two serious gaps: the induction invariant (4.1) is false as stated, and the application of Lemma 3.1 to the different-sub-block contribution uses a scale parameter that shrinks to zero, a case not covered by the lemma as stated and proved.","tokens_in":12298,"tokens_out":13990,"duration_ms":137288,"significance":"If Theorem 1.3 is correct, it is a genuinely interesting structural result: the family of weak inhomogeneous pair-correlation notions parameterized by γ are mutually independent, not merely independent of equidistribution. Theorems 1.1 and 1.2 together complete a useful diagram of implications and non-implications in the inhomogeneous setting. A notable strength is that the density construction in Theorem 1.2 is explicit and the required identities are elementary to verify. The probabilistic tools are standard, and the paper is clearly written in its overall architecture. However, the significance of the paper as a whole depends crucially on Theorem 1.3, and the current proof does not establish that theorem; the central claim is therefore not yet supported.","major_comments":[{"comment":"The displayed equality applying Lemma 3.1 to the different-sub-block term is not justified. Lemma 3.1, with N=K_r, gives concentration for the kernel at scale s/K_r^δ, whereas the term in the expansion requires scale s/N_r^δ with N_r=2rK_r. Since (K_r/N_r)^δ = (2r)^{-δ} → 0, one is applying the lemma with an effective parameter that tends to 0; Lemma 3.1 is stated and proved only for fixed s>0. The right-hand side of the displayed equality, involving the factor K_r(K_r-1)/(N_r(N_r-1)) N_r^{2-δ}, effectively assumes a concentration statement for the sub-sum over the first K_r pairs of an i.i.d. sequence of length N_r, which is not a consequence of Lemma 3.1. Without a uniform-in-s or variable-scale version of the variance estimate, the claimed limit lim_r R_Y_{N_r}(γ1,s,δ)=2s is unsupported. This is load-bearing for Theorem 1.3.","section":"§4, Step 2, display after Case 2.2"},{"comment":"The invariant (4.1) is false as written. For N=1 the prefix is (x1, x1+γ2); after step (i) of the induction one appends (x1, x1+γ2) after the whole block, producing (x1, x1+γ2, x1, x1+γ2). This is not B_2 for k=1, which would be (x1, x1, x1+γ2, x1+γ2). Thus the constructed prefix is not the concatenation B_N in the stated order. The multiplicity claim — that each xk and xk+γ2 appears exactly N times among the first M_N terms — is true, and the counting in Case 2.1 may survive if the proof is rewritten in terms of multiplicities rather than the block order explicitly assumed in 'first r positions' and 'last r positions'. But as it stands, the proof uses a false structural assertion. This must be corrected, either by altering the construction so that the invariant is genuinely satisfied or by reworking the argument to avoid the block-order claim.","section":"§4, Eq. (4.1) and the induction"},{"comment":"Lemma 3.1 is a central tool, but its proof is not self-contained: the variance estimate is asserted by reference to the case distinction in [12, Theorem 1.2] and is not proved here. Since the paper relies on this lemma both for Theorem 1.2 and (in an extended form) for Theorem 1.3, the variance bound — and the sense in which its constants are independent of s — should be stated and proved. Moreover, the 'standard approximation argument' cited from [4, p. 475] is for fixed s; the paper should either prove the variable-scale version needed in §4 or clearly restrict to fixed s and give a separate argument for the shrinking-scale application.","section":"§3, Lemma 3.1"}],"minor_comments":[{"comment":"The sentence 'there are exactly K_r such pairs' should read 'there are exactly K_r r^2 such pairs'; the lower bound later uses K_r r^2, so this is a typo.","section":"§4, Step 1"},{"comment":"The phrase 'first r positions' / 'last r positions' is inaccurate given the actual alternating construction; the counting formulas are valid for multiplicities, but the terminology should be aligned with the corrected construction.","section":"§4, Case 2.2"},{"comment":"Typo: 'This completes the poof' should be 'proof'. Also the displayed expression in Lemma 2.4 contains a typo: 'FN(t,s,N)Fδ(t−γ,s,δ)'.","section":"§3, proof of Lemma 3.1"},{"comment":"Reference [8] appears to contain a typo in the volume/pages ('Proc. Amer. Math. Soc. 7143'); please check the bibliographic data. The author list in [15] also looks inconsistent with the standard spelling.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"Theorems 1.1 and 1.2 seem solid, and the paper's topic is timely. The main concern is Theorem 1.3: the variable-scale application of Lemma 3.1 is a genuine gap, and the block invariant (4.1) is false. Both may be repairable — the multiplicity interpretation may salvage the construction, and a uniform-in-s version of Lemma 3.1 may be provable — but as submitted the proof does not establish the headline result. I recommend major revision, not rejection, because the issues are technical and local rather than showing the claims are false."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is worth reading for Theorems 1.1 and 1.2, which are cleanly proved and new. Theorem 1.3 is the advertised centerpiece, and I think it's true, but the proof as written doesn't close. The reader's report is on target about that.\n\nWhat's good: The doubled-sequence construction in Theorem 1.1 is simple and works. Theorem 1.2's density construction is explicit and verifies: the autocorrelation condition ∫g g(+γ)=1 is checked by elementary roots. These are real advances over [11,12,17,27], which only treat homogeneous weak PPC or inhomogeneous δ=1. The paper also states its limitations honestly (e.g., the open questions in Figure 2).\n\nWhere the problems are: Step 2 of Theorem 1.3 applies Lemma 3.1 to the i.i.d. sequence (x_k) at scale s/N_r^δ, which corresponds to a shrinking parameter a_r = s(2r)^{-δ} → 0. Lemma 3.1 is stated and proved for fixed s>0, and the 'standard approximation' from [4] doesn't cover a simultaneous limit with the scale tending to zero. The displayed equality with the (2s+o(1)) factor assumes a uniformity in s that the lemma doesn't give. That's a load-bearing gap. The good news is that a quantitative Borel–Cantelli argument with ε_r proportional to a_r should fix it, so I don't think the theorem is false.\n\nThe second issue is the claimed invariant (4.1). The induction as described does not produce the contiguous block B_N; the upgrade step interleaves the extra copies. For N=2 you get (x1,x1+γ2,x1,x1+γ2) instead of (x1,x1,x1+γ2,x1+γ2). This is a genuine error in the write-up, but it's minor in substance: the same-block pair counts depend only on multiplicities, and the multiplicities are correct, so the argument can be repaired by treating blocks as multisets.\n\nOne more small thing: the variance estimate in Lemma 3.1 is deferred to [12] rather than proved. That makes the paper harder to check, but it's a standard computation.\n\nOverall: Theorems 1.1 and 1.2 are publishable as is. Theorem 1.3 needs a corrected construction and a proper concentration argument at the variable scale. I'd send this to a serious referee; if the authors fix the gap, it's a good paper. The reader's reject is fair for the submitted version, but I wouldn't write the result off.","headline":"Two of the three theorems are solid and new; the third is probably true but the submitted proof has a genuine gap at the variable-scale step.","tokens_in":12826,"tokens_out":8889,"would_cite":true,"duration_ms":74773,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11K06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that weak inhomogeneous Poissonian pair correlation depends on the offset: for any two distinct offsets γ1 and γ2, a single real sequence can obey the weak pair-correlation law at γ1 and violate it at γ2.","keywords":["weak inhomogeneous Poissonian pair correlation","weak γ-PPC","equidistribution modulo one","pair correlation","independent offsets","block construction","i.i.d. sequences","number theory"],"falsifier":"Run the construction with an explicit realization of the i.i.d. base variables and compute the normalized γ1-pair count at the prefixes N_r = 2r floor(r^{δ/(1−δ)}); if for any s > 0 that count does not converge to 2s as r grows, the variable-scale application of the probabilistic lemma fails and the proof of Theorem 1.3 collapses. Independently, check the literal equality (4.1) for, say, N = 2: if the first M_2 terms do not coincide with the concatenated block B_2 in order, the stated inductive invariant is false on its face.","tokens_in":11792,"feed_emoji":"🔢","tokens_out":6896,"duration_ms":76429,"temperature":0.7,"pith_summary":"The paper studies weak inhomogeneous Poissonian pair correlation (weak γ-PPC), the normalized count of pairs in a sequence whose differences come within a shrinking distance of a fixed offset γ. It proves three separation results: weak γ-PPC does not force full γ-PPC, does not force equidistribution, and—the main discovery—for distinct offsets γ1 and γ2, a sequence can satisfy weak γ1-PPC while failing weak γ2-PPC. The construction repeats random base points in carefully sized blocks, so that pairs at offset γ2 create a permanent excess while pairs at offset γ1 follow the expected Poisson limit. A reader should care because the results show that \"weak Poissonian pair correlation\" is not a single property of a sequence but a family of mutually independent properties indexed by the shift parameter.","feed_headline":"Sequence passes weak Poisson test at one offset, not the other","feed_subtitle":"Different shifts define independent notions of pair correlation; the paper builds a sequence that separates them.","key_machinery":"The central object is the block sequence Y: at each scale r, every base point x_k is repeated r times and r copies of x_k + γ2 are appended, forming blocks. The block lengths and counts are chosen so that within-block pairs at offset γ2 act as a permanent obstruction, while cross-block pairs at offset γ1 follow the Poisson law via a probabilistic limit lemma (Lemma 3.1). That lemma states that for i.i.d. draws from a density g, the weak γ-pair correlation limit equals 2s times the autocorrelation integral of g; it supplies the quantitative control for the cross-block contributions.","core_discovery":"The paper's main theorem (Theorem 1.3) constructs, for any distinct γ1, γ2 in (0, 1/2] and any 0 < δ < 1, a real sequence Y with the following exact behavior: the normalized γ1-pair count converges to 2s for every s > 0, while the γ2-count fails to converge, staying bounded below by a positive constant along a subsequence. The construction starts from an i.i.d. uniform base sequence and repeats each point and its shift by γ2 in blocks whose length and number are tied to N^δ; the γ2 pairs inside every block force a permanent excess, while cross-block γ1 pairs reproduce the Poisson limit. The proof uses a probabilistic lemma identifying the weak pair-correlation limit for i.i.d. variables with","pith_inferences":["The same block idea suggests that, by letting the base sequence carry a non-uniform distribution, one could prescribe the limiting γ-PPC value through the autocorrelation integral, making the pair-correlation spectrum tunable; the paper does not explore this.","The construction appears robust enough to handle finite sets of offsets: one could presumably build a sequence satisfying weak γ-PPC simultaneously for several chosen offsets while failing at all offsets in a finite forbidden set, though this is not stated in the paper.","If the variable-scale extension of the probabilistic lemma can be rigorously justified, a similar technique might prove analogous independence phenomena for higher-order correlation functions modulo one."],"forward_implications":["Weak γ-PPC is not a single property of a sequence but a property of the pair (sequence, offset): changing the offset can destroy the Poissonian pair-correlation behavior.","The known relations among equidistribution, γ-PPC, and weak γ-PPC are now partially settled: weak γ-PPC neither implies γ-PPC nor equidistribution in the inhomogeneous setting.","The density construction provides a recipe for non-equidistributed sequences with prescribed weak inhomogeneous pair correlation, showing that equidistribution is not hidden inside weak inhomogeneous PPC.","The block construction gives a general principle—repeating base points can amplify one chosen offset's pair count while leaving another offset Poissonian—which can likely be reused to engineer sequences with tailored pair-correlation behavior."],"fun_headline_variants":["One sequence, two offsets, two pair-correlation verdicts","Weak Poisson test: pass at one shift, fail at another","Shifts split weak pair correlation into independent notions","Sequence passes weak-Poisson at γ1, fails at γ2"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that a probabilistic limit law, proven only for a fixed pair-distance scale, also holds when that scale shrinks to zero in the construction; if the variable-scale version fails, the claimed Poisson limit for the constructed sequence is unsupported, and the inductive step that identifies each prefix with the concatenated block must be read as a statement about multiplicities rather than literal order.","fun_headline_variants_meta":{"raw":{"variants":["One sequence, two offsets, two pair-correlation verdicts","Weak Poisson test: pass at one shift, fail at another","Shifts split weak pair correlation into independent notions","Sequence passes weak-Poisson at γ1, fails at γ2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1128,"prompt_tokens":657,"completion_tokens":471,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":401,"completion_tokens_details":{"reasoning_tokens":401}},"tokens_in":401,"tokens_out":471,"duration_ms":5545,"temperature":1.0,"reasoning_tokens":401,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T07:10:48.586581+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the construction with an explicit realization of the i.i.d. base variables and compute the normalized γ1-pair count at the prefixes N_r = 2r floor(r^{δ/(1−δ)}); if for any s > 0 that count does not converge to 2s as r grows, the variable-scale application of the probabilistic lemma fails and the proof of Theorem 1.3 collapses. Independently, check the literal equality (4.1) for, say, N = 2: if the first M_2 terms do not coincide with the concatenated block B_2 in order, the stated inductive invariant is false on its face.","supporting_citations":[],"review_version":2}