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arxiv: 2605.23884 · v1 · pith:6DKN5XHJnew · submitted 2026-05-22 · 🧮 math.FA · math-ph· math.CA· math.MP· math.SP

On almost periodicity in crystalline measures

Jan Maz\'a\v{c} , Christoph Richard , Nicolae Strungaru This is my paper

Pith reviewed 2026-05-25 02:30 UTC · model grok-4.3

classification 🧮 math.FA math-phmath.CAmath.MPmath.SP
keywords crystalline measuresalmost periodicitytranslation boundednessFourier quasicrystalsMeyer conjecturetempered distributionsRadon measuresFourier eigenmeasures
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The pith

Crystalline measures need not be almost periodic even as general distributions, since almost periodicity is equivalent to translation boundedness and counterexamples violate the latter.

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

The paper resolves two open questions on crystalline measures, which are tempered distributions whose Fourier transforms are also pure-point Radon measures of locally finite support. Meyer conjectured that all such measures are almost periodic as tempered distributions, but a counterexample already existed; Favorov then asked whether they are at least almost periodic as general distributions. The authors prove that, in the settings of Radon measures, tempered distributions, or general distributions, almost periodicity of a crystalline measure holds if and only if the measure is translation bounded. They then exhibit an explicit crystalline Fourier eigenmeasure that is not translation bounded even when viewed as a general distribution, and a second crystalline measure that is almost periodic as a tempered distribution yet fails to be a Fourier quasicrystal.

Core claim

Almost periodicity of a crystalline measure is equivalent to its translation boundedness across the three classes of Radon measures, tempered distributions, and general distributions. A crystalline Fourier eigenmeasure exists that fails translation boundedness in the space of general distributions. A further crystalline measure exists that is not a Fourier quasicrystal (in particular it is not slowly increasing) yet remains an almost periodic tempered distribution whose Fourier transform is norm almost periodic.

What carries the argument

Translation boundedness, which is shown to be equivalent to almost periodicity for crystalline measures in each of the three distributional classes.

If this is right

  • Almost periodicity holds exactly when the crystalline measure is translation bounded in the chosen class.
  • There exist crystalline measures that are not almost periodic as general distributions.
  • Crystalline measures properly contain the Fourier quasicrystals and can exhibit almost periodicity without slow increase.
  • The boundary between translation-bounded and non-translation-bounded crystalline measures is now sharply delineated.

Where Pith is reading between the lines

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

  • Similar translation-boundedness characterizations may apply to other classes of measures with pure-point Fourier transforms.
  • The constructions supply concrete test cases for studying growth conditions or regularity properties beyond the crystalline definition.
  • Questions about almost periodicity for related objects such as tempered distributions with discrete spectra could be settled by analogous boundedness criteria.

Load-bearing premise

The constructed objects remain crystalline measures (both the object and its Fourier transform are pure-point Radon measures of locally finite support) while failing to be translation bounded as general distributions.

What would settle it

An explicit verification that the constructed Fourier eigenmeasure is in fact translation bounded when tested against all compactly supported continuous test functions, or a proof that any object satisfying the crystalline definition must automatically be translation bounded.

read the original abstract

Meyer defined crystalline measures as tempered distributions $\mu$ such that both $\mu$ and its Fourier transform $\widehat\mu$ are pure-point Radon measures of locally finite support. He conjectured that every crystalline measure is almost periodic as a tempered distribution. Favorov constructed a counterexample and asked whether crystalline measures are at least almost periodic as general distributions. To resolve Favorov's question, we first show that the almost periodicity of a crystalline measure is characterised in terms of its translation boundedness, in any class of Radon measures, tempered distributions, or general distributions. We then construct a crystalline Fourier eigenmeasure that fails to be translation bounded even as a distribution. We finally construct a crystalline measure that fails to be a~Fourier quasicrystal (in particular, it fails to be slowly increasing), but it is an almost periodic tempered distribution whose Fourier transform is even a norm almost periodic measure. Our examples fully resolve the questions of Meyer and Favorov and sharply delineate the class boundary of translation boundedness. They also demonstrate the unusual behaviour of crystalline measures beyond the class of Fourier quasicrystals.

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

2 major / 1 minor

Summary. The paper characterizes the almost periodicity of crystalline measures (tempered distributions μ such that both μ and ˆμ are pure-point Radon measures with locally finite support) in terms of translation boundedness, holding in the classes of Radon measures, tempered distributions, and general distributions. It constructs a crystalline Fourier eigenmeasure that fails to be translation bounded even as a distribution, and a second crystalline measure that is almost periodic as a tempered distribution (with norm almost periodic Fourier transform) but fails to be a Fourier quasicrystal or slowly increasing. These resolve the questions of Meyer and Favorov.

Significance. If the constructions are valid, the results resolve open questions on almost periodicity for crystalline measures by providing an explicit negative answer to Favorov via a Fourier eigenmeasure counterexample and by delineating the precise role of translation boundedness. The characterization theorem is a clean structural result, and the examples demonstrate behavior strictly outside the Fourier quasicrystal class while remaining almost periodic in the tempered sense. Explicit constructions of this type are a strength.

major comments (2)
  1. [§4] §4 (Constructions), first example: the verification that the constructed Fourier eigenmeasure μ satisfies both μ and ˆμ being pure-point Radon measures of locally finite support (while ˆμ fails to be translation bounded as a distribution) is not load-bearing if left implicit; the text must explicitly confirm the support and Radon conditions hold without forcing translation boundedness, as this is the central counterexample to Favorov's question.
  2. [§3] Characterization theorem (likely §3): the equivalence between almost periodicity and translation boundedness is stated for general distributions, but the proof sketch does not address whether the crystalline assumption (pure-point Radon with locally finite support) is used in both directions or only one; this affects whether the characterization applies uniformly across the three classes mentioned in the abstract.
minor comments (1)
  1. [Introduction] Notation for the Fourier transform and the spaces (tempered vs. general distributions) should be fixed consistently in the introduction and §2 to avoid ambiguity when stating the characterization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major comment below and will make the requested revisions to improve clarity and explicitness.

read point-by-point responses

  1. Referee: [§4] §4 (Constructions), first example: the verification that the constructed Fourier eigenmeasure μ satisfies both μ and ˆμ being pure-point Radon measures of locally finite support (while ˆμ fails to be translation bounded as a distribution) is not load-bearing if left implicit; the text must explicitly confirm the support and Radon conditions hold without forcing translation boundedness, as this is the central counterexample to Favorov's question.

    Authors: We agree that the verification must be made fully explicit rather than left implicit. In the revised version of §4, we will add a dedicated paragraph confirming that the constructed μ and ˆμ are pure-point Radon measures with locally finite support, while separately verifying that translation boundedness fails as a distribution. This will be done without altering the construction itself. revision: yes

  2. Referee: [§3] Characterization theorem (likely §3): the equivalence between almost periodicity and translation boundedness is stated for general distributions, but the proof sketch does not address whether the crystalline assumption (pure-point Radon with locally finite support) is used in both directions or only one; this affects whether the characterization applies uniformly across the three classes mentioned in the abstract.

    Authors: The theorem states the equivalence separately within each of the three classes (Radon measures, tempered distributions, general distributions) for crystalline measures. The crystalline assumption is invoked to guarantee that the objects remain in the respective class, but the direction 'almost periodic implies translation bounded' holds by the definition of almost periodicity alone, while the converse direction uses the pure-point and locally finite support properties to construct the almost periodic function. We will expand the proof sketch in the revised §3 to explicitly separate the two directions and confirm that the result applies uniformly in each class as stated in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions and characterization are independent of inputs

full rationale

The paper's central results consist of a characterization of almost periodicity for crystalline measures in terms of translation boundedness, followed by explicit constructions of counterexamples (a crystalline Fourier eigenmeasure that is not translation bounded as a distribution, and a crystalline measure that is not a Fourier quasicrystal). These steps rely on direct verification against the definitions of crystalline measures (pure-point Radon measures of locally finite support for both μ and ˆμ) and do not reduce any claimed prediction or theorem to a fitted parameter, self-citation chain, or ansatz smuggled from prior work. The characterization is derived from standard properties of Radon measures and distributions rather than being forced by the constructions themselves. No load-bearing step equates to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; relies on standard properties of Fourier transforms on distributions and Radon measures with no free parameters or invented entities mentioned.

axioms (1)
  • domain assumption Fourier transform extends to tempered distributions and general distributions while preserving the pure-point Radon measure property under the crystalline definition
    Invoked throughout the definition of crystalline measures and the constructions described in the abstract.

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