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Pair correlation of $\alpha n^{\theta}$ for random $\theta$

T0 review · 1 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read The sequence {α n^θ} has Poissonian pair correlation for Lebesgue-almost all θ in (0, 3/5) ∪ (3, ∞).

desk verdict 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. read the letter →

arxiv 2606.03781 v1 pith:GVI4BRRJ submitted 2026-06-02 math.NT

Andrei Shubin This is my paper · ORCID
classification math.NT
keywords paircorrelationPoissoniandistributionfractionalpartsDiophantineapproximationRiemannzetafunctionexponentpairsnumbertheory
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

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.

What carries the argument

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.

What would settle it

A positive-measure set of θ inside (0, 3/5) for which the pair-correlation statistic fails to approach the Poisson limit.

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Extended reading notes

Core claim

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,∞).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

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.

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 (1)
  1. [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.
minor comments (2)
  1. [§3 (splitting argument)] 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.
  2. [counting estimates] The dependence of the implied constants on α should be tracked through the estimates, even if α is fixed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: [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.

    Authors: 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: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation reduces to independent zeta-moment and exponent-pair bounds

full rationale

The paper splits the θ-integral over short intervals, bounds most contributions via the first derivative of the phase, and controls the exceptional set E via Diophantine counting estimates. These estimates are proved using moments of the Riemann zeta function and exponent pairs, which are classical external results (independent of the present claim and not obtained via self-citation). The approach differs from the cited Technau-Yesha repulsion principle and does not reduce any prediction or central statement to a fitted input or self-referential definition. No load-bearing self-citations, ansatz smuggling, or renaming of known results occur. The derivation is self-contained against external benchmarks.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all technical ingredients are referenced to prior literature.

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Cite this review

Pith. "Pith review of Pair correlation of $\alpha n^{\theta}$ for random $\theta$." pith.science (2026). https://pith.science/paper/GVI4BRRJ

@misc{pith2026260603781,
  author       = {Pith},
  title        = {Pith review of: Pair correlation of $\alpha n^\theta$ for random $\theta$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GVI4BRRJ}},
  note         = {Machine review of arXiv:2606.03781}
}
abstract

For fixed $\alpha>0$, we show that the sequence $\{\alpha n^{\theta}\}$ has Poissonian pair correlation for Lebesgue-almost all $\theta \in (0,\frac{3}{5})\cup(3,\infty)$. This improves a result of Technau and Yesha, who proved the same for almost all $\theta>7$. The approach of Technau and Yesha was based on a repulsion principle, which roughly allows one to estimate the variance of the pair correlation function using the fourth derivative of the phase. In our approach, we split the $\theta$-integration in the variance into many short intervals and show that most of the integrals can be estimated using the first derivative. The problem is then reduced to several counting estimates, which we prove using moments of the Riemann zeta function and exponent pairs.

Discussion (0). Continue with ORCID to comment.

Reference graph

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