Pure-point diffraction conjecture for visible points of irreducible cut-and-project sets

Let d≥2d\geq 2 and let Λ=Λ(W,L)⊂Rd\Lambda=\Lambda(\mathcal{W},\mathcal{L})\subset\mathbb{R}^d be an irreducible cut-and-project set. Let A⊂Rd\mathcal{A}\subset\mathbb{R}^d be a van Hove sequence, and suppose that the density of the visible points Λvis\Lambda_{\mathrm{vis}} along A\mathcal{A} exists and is positive. Pure-point diffraction conjecture for visible points. The diffraction spectrum

γΛvis^\widehat{\gamma_{\Lambda_{\mathrm{vis}}}}

is a translation bounded pure point measure. This would extend the almost-everywhere result for random cut-and-project sets to every irreducible cut-and-project set satisfying the stated positive-density hypothesis. The conjecture concerns the diffraction of visible points and remains open in the generality stated.

References

Primary source

Rishi Kumar and Carlos Ospina, “On the diffraction spectrum of the set of visible points in lattices and certain cut-and-project sets”, arXiv:2606.30132 (2026).

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