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arxiv: 2606.30132 · v1 · pith:IZXL7ZLNnew · submitted 2026-06-29 · 🧮 math.NT · math-ph· math.MP

On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets

Rishi Kumar , Carlos Ospina This is my paper

Pith reviewed 2026-06-30 05:20 UTC · model grok-4.3

classification 🧮 math.NT math-phmath.MP
keywords diffraction spectrumvisible lattice pointscut-and-project setspure point spectrumtranslation bounded measuresharmonic analysissimultaneous visibility
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The pith

Points visible from several lattice points at once have a pure point diffraction spectrum with explicit coefficients.

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

The paper starts from the established fact that lattice points visible from a single origin in Z^k possess a translation-bounded pure point diffraction spectrum. It proves the same spectral property persists when a point must be visible from every member of a finite collection of lattice points, and derives closed-form expressions for the diffraction coefficients in that case. The method is then applied to show that visible points from the origin in selected cut-and-project sets likewise exhibit translation-bounded pure point diffraction. A reader cares because these constructions appear in models of ordered but non-periodic structures whose long-range correlations are encoded precisely in the diffraction measure.

Core claim

The set of points in Z^k simultaneously visible from a finite collection of lattice points admits a translation-bounded pure point diffraction spectrum whose coefficients are given by explicit formulas; the identical spectral conclusion holds for the visible points from the origin inside certain cut-and-project sets.

What carries the argument

The simultaneous visibility condition with respect to a finite set of origins, which is shown to preserve the uniformity and density properties that permit the diffraction measure to be recovered by the same harmonic-analysis methods used for the single-origin case.

If this is right

  • Explicit coefficient formulas become available for any finite collection of origins in any dimension k greater than or equal to 2.
  • The diffraction spectrum remains pure point and translation bounded when the same visibility notion is transferred to certain cut-and-project sets.
  • The support of the diffraction measure is determined by the common visible lattice points and can be described without reference to the particular choice of origins beyond their finiteness.

Where Pith is reading between the lines

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

  • The same visibility construction may supply new examples of model sets whose diffraction can be computed directly rather than via approximation.
  • If the explicit formulas extend to infinite but periodic collections of origins, the result could classify diffraction spectra for a broader family of point sets arising in quasicrystal models.

Load-bearing premise

Simultaneous visibility from a finite number of origins preserves the uniformity and density properties that allow the diffraction measure to be analyzed by the existing harmonic-analysis techniques.

What would settle it

An explicit computation or numerical approximation of the diffraction measure for a concrete finite set of origins that reveals a non-zero continuous component would falsify the pure-point claim.

Figures

Figures reproduced from arXiv: 2606.30132 by Carlos Ospina, Rishi Kumar.

Figure 1
Figure 1. Figure 1: Diffraction of the visible points of Z 2 [BG13, p. 424]. Question. What can be said about the diffraction spectrum of the points in Z k that are simultaneously visible from every element of S? Denote by P the set of prime numbers in N. For each p P P, the map πp denotes the natural projection (1) πp : Z k ÝÑ pZ{pZq k , px1, . . . , xkq ÞÝÑ ` x1 mod p, . . . , xk mod p ˘ , and sppq def “ # pπppSqq is the ca… view at source ↗
Figure 2
Figure 2. Figure 2: Ammann–Beenker point set (left) and its diffraction image (right), adapted from [Sin] and [BG13, p. 372], respectively. On the other hand, Hammarhjelm [Ham22] constructed examples of cut-and-project sets ΛpW,Lq Ď R 2 for which the inequality in (4) is strict and computed θpΛvisq explicitly. The examples considered by Hammarhjelm included the Amman-Beenker point set and sets associated with the Penrose tili… view at source ↗
read the original abstract

Let $k\geq 2$ be a positive integer. It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum. We investigate these properties for sets of points simultaneously visible from a finite set of lattice points $ \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \subseteq \mathbb{Z}^k$. We provide explicit formulas for the coefficients of the diffraction spectrum. Additionally, we generalize our procedure to show that the set of visible points from the origin in certain classes of cut-and-project sets has a translation bounded pure point diffraction spectrum.

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 / 2 minor

Summary. The manuscript extends the known result that visible lattice points from the origin in Z^k possess a translation-bounded pure-point diffraction spectrum to the multi-origin case of points simultaneously visible from a finite set {x1,...,xn} subset Z^k, supplying explicit formulas for the diffraction coefficients; it further generalizes the argument to show that visible points from the origin in certain cut-and-project sets likewise have translation-bounded pure-point diffraction spectra.

Significance. The explicit formulas constitute a concrete, verifiable strengthening of the single-origin theory and supply the concrete evidence that the multi-origin visibility condition preserves the uniformity and density properties required for the harmonic-analysis argument. The cut-and-project generalization demonstrates that the visibility construction is robust beyond the lattice setting, which is of direct interest to the study of model sets and diffraction in aperiodic order.

minor comments (2)
  1. [Abstract] The abstract states that explicit formulas are provided but does not indicate their form (e.g., whether they involve sums over characters or Möbius-type factors); a single illustrative expression would improve immediate readability.
  2. [Introduction] Notation for the finite set of origins and the simultaneous-visibility condition should be introduced with a displayed definition early in the introduction to avoid ambiguity when the multi-origin case is first stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of the explicit formulas as a strengthening of prior results, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends a known result (single-origin visible points in Z^k have translation-bounded pure-point diffraction) to the multi-origin case and to certain cut-and-project sets, supplying explicit coefficient formulas. The abstract and described claims contain no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations; the central claims rest on independent harmonic-analysis techniques applied to the new visibility conditions. This is the normal non-circular outcome for an extension paper whose base case is externally established.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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discussion (0)

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Reference graph

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