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arxiv: 2604.13310 · v1 · submitted 2026-04-14 · 🧮 math.FA

Higher-Order Autocorrelations on Finite Abelian Groups

Aaron Agulnick , Toby Busick-Warner This is my paper

Pith reviewed 2026-05-10 13:26 UTC · model grok-4.3

classification 🧮 math.FA
keywords higher-order autocorrelationsfinite abelian groupsFourier transform vanishingsignal determinationautocorrelation boundsZ6 classificationphase retrieval
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The pith

Knowledge of where a signal's Fourier transform vanishes yields two new upper bounds on the autocorrelation order needed to determine the signal on any finite abelian group.

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

The paper proves two explicit upper bounds on the minimal order of autocorrelation data sufficient to uniquely recover a signal, when the set of zeros of its Fourier transform is known in advance. This matters because applications such as crystallography, image processing, and communications often face the problem of reconstructing a signal from partial autocorrelation measurements, and prior knowledge of Fourier support can reduce the data volume required. The bounds apply uniformly to all finite abelian groups, including those that are not cyclic. The authors complement the upper bounds by classifying all signals on the group Z6 that fail to be determined by fifth-order autocorrelations and by exhibiting analogous examples on Z30.

Core claim

We prove two new upper bounds on the order of data needed to determine a signal on a general (i.e., not necessarily cyclic) finite abelian group depending on some knowledge of the vanishing of the signal's Fourier transform. In investigating lower bounds on the required data, we classify signals on Z6 not determined by their fifth-order data and provide analogous examples on Z30.

What carries the argument

The vanishing set of the Fourier transform, which supplies the information needed to derive explicit upper bounds on the autocorrelation order sufficient for unique signal recovery.

If this is right

  • When the Fourier transform vanishes on a large set, the autocorrelation order needed for reconstruction drops accordingly.
  • The bounds apply equally to cyclic and non-cyclic finite abelian groups.
  • On Z6, the signals not recovered by fifth-order data are completely classified.
  • Analogous undetermined signals exist on Z30.

Where Pith is reading between the lines

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

  • Incorporating known Fourier support could reduce computational cost in phase-retrieval pipelines that already compute autocorrelations.
  • The classification on small groups may indicate which algebraic features force higher-order data in general.
  • Similar vanishing-set arguments might extend the bounds to other classes of groups or to continuous settings.

Load-bearing premise

The vanishing set of the signal's Fourier transform is known and can be leveraged to obtain the stated upper bounds.

What would settle it

A concrete signal on some finite abelian group whose Fourier transform vanishes on a given set, yet requires strictly higher-order autocorrelation data than the claimed bound to achieve unique determination.

read the original abstract

The question of determining a signal from its higher-order autocorrelation data is of practical interest in fields as varied as X-ray crystallography, image processing, and satellite communications. At the heart of the issue is how much of this autocorrelation data one truly needs. We prove two new upper bounds on the order of data needed to determine a signal on a general (i.e. not necessarily cyclic) finite abelian group depending on some knowledge of the vanishing of the signal's Fourier transform. In investigating lower bounds on the required data, we classify signals on $\mathbb{Z}_6$ not determined by their fifth-order data and provide analogous examples on $\mathbb{Z}_{30}$.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript proves two new upper bounds on the minimal order of autocorrelation data sufficient to uniquely determine a signal on a general finite abelian group G, conditioned on knowledge of the zero set of its Fourier transform. It additionally classifies signals on the cyclic group Z_6 that are not determined by fifth-order autocorrelations and supplies analogous examples on Z_30.

Significance. If the bounds hold, the work advances phase-retrieval and signal-recovery theory by showing how Fourier-support information can reduce the required autocorrelation order on non-cyclic groups, with direct relevance to crystallography, image processing, and communications. The explicit classifications on Z_6 and Z_30 supply concrete, falsifiable examples that test the sharpness of the bounds and illustrate undetermined signals.

minor comments (3)
  1. [Introduction] The introduction would benefit from an explicit statement of the two upper bounds (e.g., as Theorem 1 and Theorem 2) immediately after the abstract, including the precise dependence on the size of the Fourier zero set.
  2. Notation for the k-th order autocorrelation function and its Fourier transform should be introduced once and used consistently; occasional shifts between capital and lower-case letters for the same object appear in the cyclic-group sections.
  3. [Z_6 and Z_30 classifications] The classification tables or lists for Z_6 and Z_30 would be clearer if accompanied by a brief remark on how the undetermined signals were enumerated (exhaustive search versus algebraic argument).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in phase-retrieval and signal-recovery theory, and recommendation for minor revision. No specific major comments or requested changes were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; bounds derived via standard Fourier analysis on finite abelian groups

full rationale

The paper's central results consist of two conditional upper bounds on the minimal autocorrelation order sufficient to recover a signal on a general finite abelian group G, given knowledge of the zero set of its Fourier transform. These bounds are obtained by leveraging the support of the Fourier transform to reduce effective degrees of freedom, with explicit classification of undetermined signals provided for the cyclic cases Z_6 and Z_30. The derivation rests on the standard theory of Fourier analysis over finite abelian groups (Pontryagin duality, character sums, and support constraints) without any self-referential fitting, parameter estimation from the target data, or load-bearing self-citations. No step reduces by construction to its own inputs, and the classification examples are direct verifications rather than predictions forced by prior fits. The work is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest entirely on standard properties of Fourier transforms and characters on finite abelian groups; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (1)
  • standard math Fourier transform on finite abelian groups satisfies the standard inversion and convolution theorems
    Invoked to relate autocorrelation data to the Fourier side and to use vanishing information.

pith-pipeline@v0.9.0 · 5401 in / 1049 out tokens · 29676 ms · 2026-05-10T13:26:35.998968+00:00 · methodology

discussion (0)

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Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    A note on translation in- variants

    [AK62] Roy L. Adler and Alan G. Konheim. “A note on translation in- variants”. In:Proc. Amer. Math. Soc.13 (1962), pp. 425–428.issn: 0002-9939,1088-6826.doi: 10.2307/2034951 .url: https://doi. org/10.2307/2034951. [CO25] W. Riley Casper and Bobby Orozco. “Reconstructing Gridded Data fromHigherAutocorrelations”.In:(2025).arXiv: 2503.21022 [cs.CV]. url:http...

    work page doi:10.2307/2034951 1962

  2. [2]

    The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets

    Chap. 8: Quadratic Integer Rings.url:https://dummit. cos.northeastern.edu/teaching_sp24_4527/numthy_8_quadratic_ integer_rings_v3.50.pdf. [GM95] F. Alberto Grünbaum and Calvin C. Moore. “The use of higher-order invariants in the determination of generalized Patterson cyclotomic sets”. In:Acta Cryst. Sect. A51.3 (1995), pp. 310–323.issn: 0108- 7673.doi: 10...

    work page doi:10.1107/s0108767394009827.url: 1995

  3. [3]

    Patterson and Pattersons. Fifty Years of the Patterson Function. Jenny P. Glusker , Betty K. Patterson , Miriam Rossi

    issn: 0034-4885.doi: 10.1088/0034-4885/54/11/002.url: https: / / dx . doi . org / 10 . 1088 / 0034 - 4885 / 54 / 11 / 002(visited on 10/14/2024). [Moo89] Paul Brian Moore. “Patterson and Pattersons. Fifty Years of the Patterson Function. Jenny P. Glusker , Betty K. Patterson , Miriam Rossi”. In:The Journal of Geology97.3 (May 1989). Publisher: The Univers...

    work page doi:10.1088/0034-4885/54/11/002.url: 2024