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REVIEW 3 major objections 4 minor 30 references

Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.10773 v2 pith:YLPU4TU4 submitted 2026-07-12 math.NT math.PR

Zhiqin Tang , Qing-Long Zhou This is my paper
classification math.NTmath.PR MSC 11K06
keywords weakinhomogeneousPoissonianpaircorrelationγ-PPCequidistributionmodulooneindependentoffsetsblockconstructioni.i.d.sequencesnumbertheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§4, Step 2, display after Case 2.2] 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.
  2. [§4, Eq. (4.1) and the induction] 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.
  3. [§3, Lemma 3.1] 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.
minor comments (4)
  1. [§4, Step 1] 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.
  2. [§4, Case 2.2] 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.
  3. [§3, proof of Lemma 3.1] 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)'.
  4. [References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constructions are explicit probabilistic counterexamples proved from first principles; cited prior work is used as lemmas, not as the target conclusion.

full rationale

The paper's derivation chain does not reduce to its inputs. Theorem 1.1 constructs a random sequence x_{2n-1}=y_n, x_{2n}=y_n+γ and proves weak γ-PPC almost surely via an integral characterization (Lemma 2.1) and uniform convergence of local counting functions (Lemma 2.3), none of which assumes the target. Theorem 1.2 obtains a density g with ∫g(x)g(x+γ)dx=1 and g not identically 1; Lemma 3.1 is proved independently by expectation/variance for i.i.d. sequences with density g, so the conclusion R_N→2s follows from a verified limit computation, not from an assumed value. Theorem 1.3 constructs a block sequence and proves the two PPC properties separately: failure of weak γ2-PPC is a direct lower-bound estimate, and weak γ1-PPC is derived by applying Lemma 3.1 to the auxiliary i.i.d. uniforms (x_k) with shifts γ1, γ1−γ2, γ1+γ2. This application is a genuine reduction to an independent theorem, not a restatement of the desired property. The citations to [11], [12], and [4] are to other authors' work used as tools (an integral characterization, a variance computation pattern, and a standard subsequence-to-full approximation), and no load-bearing premise is justified solely by a self-citation. Concerns raised by the skeptic about the variable-scale application of Lemma 3.1 or the block invariant (4.1) are potential correctness or technical gaps, not circularity: even if those gaps were fatal, the argument would be incomplete rather than self-referential. There are no fitted parameters renamed as predictions and no uniqueness theorem invoked by the authors to force a choice. The central claims have independent content and are derived from explicit probabilistic constructions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants or new entities. Its dependence is on standard probability tools (Bernstein, Borel–Cantelli) and on borrowed technical lemmas from [11,12] whose proofs are not fully reproduced. The most fragile borrowed result is the variance estimate and the variable-scale extension of Lemma 3.1.

assumptions (4)
  • domain assumption Lemma 2.1: weak γ-PPC is equivalent to the integral condition I_N(γ,s,δ)→s².
    The proof is not given; the paper says it follows the same lines as [11, Lemma 9]. This equivalence underlies Theorem 1.1.
  • domain assumption Variance estimate in Lemma 3.1: Var(R_N) ≪ (1/N^{2−δ}) E[R_N].
    The computation is deferred to [12, Theorem 1.2] (pp. 6–7) and not reproduced for the δ∈(0,1) case. Lemma 3.1 is used in both Theorem 1.2 and Theorem 1.3.
  • domain assumption Standard approximation: subsequence convergence (N_m=m² or N_r) implies full-sequence convergence when consecutive indices have ratio →1.
    Invoked in Lemma 3.1 and Theorem 1.3 Step 2, citing [4, p.475]. The argument is not written out.
  • ad hoc to paper Lemma 3.1 extends to windows whose effective parameter s′→0.
    In Theorem 1.3 Step 2, the different-sub-block sums use scale s/N_r^δ, which for the i.i.d. sequence (x_k) corresponds to s′ = s/(2r)^δ → 0. This is an unproved strengthening of Lemma 3.1.

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Cite this review

Pith. "Pith review of Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution." pith.science (2026). https://pith.science/paper/YLPU4TU4

@misc{pith2026260710773,
  author       = {Pith},
  title        = {Pith review of: Weak Inhomogeneous Poissonian Pair Correlation and Equidistribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YLPU4TU4}},
  note         = {Machine review of arXiv:2607.10773}
}
abstract

This paper investigates inhomogeneous Poissonian pair correlation (PPC), its weak form, and equidistribution. We establish that weak inhomogeneous PPC does not imply inhomogeneous PPC. Furthermore, we construct a sequence satisfying weak inhomogeneous PPC that fails to be equidistributed, which stands in sharp contrast to the homogeneous case. Finally, we prove that for distinct $\gamma_1, \gamma_2 \in (0, \frac{1}{2}]$, weak $\gamma_1$-PPC does not imply weak $\gamma_2$-PPC, showing that different inhomogeneous parameters give rise to mutually independent notions of weak inhomogeneous PPC.

Figures

Figures reproduced from arXiv: 2607.10773 by the authors.

Figure 1
Figure 1. Relations among equidistribution, PPC and weak PPC [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Relations among equidistribution, γ-PPC and weak γ-PPC 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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