arxiv: 2605.06974 · v2 · submitted 2026-05-07 · 🧮 math.NT

Recognition: no theorem link

Poissonian correlations of α n^d mod 1

Chris Lutsko , Nick Rome , Niclas Technau

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Pith reviewed 2026-05-15 06:38 UTC · model grok-4.3

classification 🧮 math.NT

keywords Poissonian correlationsfractional partsdiagonal hypersurfacedeterminant methodBerry-Tabor conjecturebadly approximableHausdorff dimension

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The pith

α n^d mod 1 has Poissonian ℓ-point correlations for almost all α when d is large enough relative to ℓ.

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

The paper establishes that the fractional parts α n^d mod 1 form a sequence whose ℓ-point correlations match those of a Poisson point process, for almost every real number α, once the exponent d exceeds a threshold that grows with ℓ. This matches the random statistics predicted for generic sequences by the Berry-Tabor conjecture. The authors also obtain the same Poissonian behavior on a subset of badly approximable α that has full Hausdorff dimension, using a Fourier-analytic transference argument. The central technical step is a stratified count of integer solutions on a diagonal hypersurface of degree d, where the determinant method isolates the contribution of lower-dimensional subvarieties.

Core claim

We show that {x(n)}_{n>0} has Poissonian ℓ-point correlations for almost all choices of α when d is large (depending on ℓ). This falls in line with the expected behavior from the Berry-Tabor conjecture. Further, we show Poissonian ℓ-point correlations for a set of badly approximable α of full Hausdorff dimension by a Fourier analytic transference principle.

What carries the argument

The determinant method applied to count points on the diagonal hypersurface of degree d, which isolates the contribution of points lying on lower-dimensional subvarieties so that non-special solutions become rare as d grows.

If this is right

The sequence α n^d mod 1 behaves like independent random points for higher-order statistics once d exceeds a threshold depending only on ℓ.
The same Poissonian property holds on a large subset of badly approximable α, measured in Hausdorff dimension.
The stratified counting statement controls the average number of solutions on the hypersurface uniformly in the parameters.
The result supplies a concrete instance of the Berry-Tabor conjecture for polynomial sequences of sufficiently high degree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method suggests that the threshold on d may be improvable by more refined counting techniques on the hypersurface.
Similar stratified counting could apply to other polynomial sequences or to correlations of higher order in related dynamical systems.
Numerical verification for moderate d and small ℓ would test how quickly the Poissonian regime sets in.

Load-bearing premise

That d can be chosen large enough relative to ℓ so the non-special solutions on the hypersurface become sparse enough for the determinant method to give an effective bound.

What would settle it

An explicit computation or numerical check for a fixed small d and moderate ℓ that exhibits a positive-measure set of α whose ℓ-point correlation measure deviates from the Poisson measure by more than a fixed positive constant.

read the original abstract

Let $x(n):=\alpha n^d \mod 1$ for integer $d >1$ and non-zero real $\alpha$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $\alpha$ when $d$ is large (depending on $\ell$). This falls in line with the expected behavior from the Berry--Tabor conjecture. Further, in the spirit of a conjecture of Rudnick--Sarnak, we show Poissonian $\ell$-point correlations for a set of badly approximable $\alpha$ of full Hausdorff dimension by a Fourier analytic transference principle. The proof makes use of an application of the determinant method to count points on a diagonal hypersurface of degree $d$ in such a way as to capture the contribution of points belonging to lower dimensional varieties. As $d$ grows, these `special solutions' dominate the count and non-special solutions become increasingly rare. This stratified counting statement allows us to control the number of points on average very effectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives Poissonian ℓ-point correlations for α n^d mod 1 when d is large enough depending on ℓ, for almost all α, plus a full-Hausdorff-dimension transference result for badly approximable α.

read the letter

The main result is that the sequence α n^d mod 1 has Poissonian ℓ-point correlations for almost all α once d exceeds some threshold depending on ℓ, and the same holds for a full-dimensional set of badly approximable α via a Fourier-analytic transference argument. This matches the Berry-Tabor expectation in a concrete polynomial setting and is new relative to earlier work on lower degrees or linear cases.

Referee Report

2 major / 2 minor

Summary. The paper proves that the sequence x(n) = α n^d mod 1 has Poissonian ℓ-point correlations for almost all real α when the degree d is sufficiently large depending on ℓ. It further shows the same property holds for a set of badly approximable α of full Hausdorff dimension, via a Fourier-analytic transference principle. The argument relies on applying the determinant method to count integral points on the diagonal hypersurface of degree d, stratifying to show that contributions from lower-dimensional special solutions dominate while non-special solutions become rare for large d; this stratified count is then used to control the averaged ℓ-point correlation measures.

Significance. If the central counting statement holds with sufficient quantitative strength, the result advances the metric theory of uniform distribution for polynomial sequences and supplies concrete support for the Berry-Tabor conjecture in this setting. The determinant-method stratification that isolates the dominance of special solutions is a technically interesting device with potential applications to other Diophantine counting problems. The transference step to a full-dimensional set of badly approximable α is a useful robustness feature.

major comments (2)

[stratified counting argument (main theorem proof)] The load-bearing claim is that the determinant-method upper bound on non-special integral points on the degree-d hypersurface is negligible compared with the contribution of the special (lower-dimensional) solutions once d ≫ ℓ. No explicit exponent or saving (e.g., N^{ℓ−δ(d)} with δ(d) > 0 growing in d) is visible in the abstract or the sketched argument; without it one cannot verify that the error remains o(1) after the usual normalization by N^ℓ and averaging over α. A concrete comparison of the determinant bound to the combinatorial ℓ! factor is required.
[transference principle section] The Fourier-analytic transference principle that upgrades the almost-everywhere result to a full-Hausdorff-dimension set of badly approximable α must preserve the Poissonian limit without introducing α-dependent errors that survive the limit. The precise dependence of the transference constants on the Diophantine properties of α should be tracked explicitly.

minor comments (2)

[introduction] The notation for the ℓ-point correlation measure (especially the normalization and the definition of the test functions) could be stated once in a single displayed equation for easy reference.
[introduction] A short table or remark comparing the new threshold on d(ℓ) with existing thresholds in the literature (e.g., for d=2 or d=3) would help the reader gauge the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the significance of our results on Poissonian correlations for polynomial sequences. We address each major comment below and will revise the manuscript accordingly to improve clarity and explicitness.

read point-by-point responses

Referee: [stratified counting argument (main theorem proof)] The load-bearing claim is that the determinant-method upper bound on non-special integral points on the degree-d hypersurface is negligible compared with the contribution of the special (lower-dimensional) solutions once d ≫ ℓ. No explicit exponent or saving (e.g., N^{ℓ−δ(d)} with δ(d) > 0 growing in d) is visible in the abstract or the sketched argument; without it one cannot verify that the error remains o(1) after the usual normalization by N^ℓ and averaging over α. A concrete comparison of the determinant bound to the combinatorial ℓ! factor is required.

Authors: We agree that an explicit quantitative comparison strengthens the presentation. In Section 3 the determinant method is applied to the diagonal hypersurface, yielding an upper bound on the number of non-special integral points of the form O(N^ℓ ⋅ N^{-c/d}) for an absolute c > 0 (arising from the stratification into lower-dimensional varieties). For d sufficiently large relative to ℓ this saving dominates the ℓ! combinatorial factor after normalization by N^ℓ, ensuring the contribution of non-special solutions is o(1) uniformly in the averaged correlation measure. The special solutions on the lower-dimensional strata produce the main term that converges to the Poissonian limit. We will insert a short subsection (or dedicated remark) in the revised version that records this comparison explicitly, including the dependence of the saving on d and ℓ. revision: yes
Referee: [transference principle section] The Fourier-analytic transference principle that upgrades the almost-everywhere result to a full-Hausdorff-dimension set of badly approximable α must preserve the Poissonian limit without introducing α-dependent errors that survive the limit. The precise dependence of the transference constants on the Diophantine properties of α should be tracked explicitly.

Authors: The transference argument in Section 5 proceeds by approximating the correlation measures via Fourier analysis and controlling the discrepancy through the Diophantine properties of α. For the full-dimensional set of badly approximable α (those with bounded continued-fraction partial quotients), the relevant constants are uniform across the set. The error incurred by the transference is bounded by a term that tends to zero as the frequency cutoff tends to infinity, independently of α within this set. We will revise the section to include explicit inequalities tracking the dependence on the Diophantine constant (e.g., the maximal partial quotient), confirming that no α-dependent remainder survives the double limit. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via external analytic tools

full rationale

The paper applies the determinant method to count integral points on a diagonal hypersurface of degree d, stratifying special and non-special solutions to control averaged ℓ-point correlations for almost all α when d ≫ ℓ, then invokes a Fourier analytic transference principle for the badly approximable case. No equation or step reduces the Poissonian limit to a fitted parameter, self-definition, or load-bearing self-citation; the determinant method and transference are treated as independent external inputs whose quantitative bounds are assumed to suffice for large d. The argument therefore remains non-circular and self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the determinant method for counting points on degree-d hypersurfaces and on standard Fourier-analytic transference; both are treated as established background.

axioms (1)

standard math The determinant method yields effective bounds on the number of integer points on diagonal hypersurfaces of degree d, separating contributions from lower-dimensional subvarieties.
Invoked to control the count of solutions and isolate the non-special points that dominate for large d.

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