Intrinsic Characterizations of Regular Model Sets

Definitions and mathematical statement

The source requests an intrinsic, simple characterization of regular model sets. In the standard Euclidean formulation, a Delone set $\Lambda\subset{\mathbb{R}}^d$ is uniformly discrete and relatively dense: distinct points have a uniform positive separation, and every sufficiently large ball meets $\Lambda$.

A cut-and-project scheme consists of ${\mathbb{R}}^d$, a locally compact abelian internal group $H$, and a lattice $\mathcal L\subset{\mathbb{R}}^d\times H$, with the physical projection $\pi_1|_{\mathcal L}$ injective and the internal projection $\pi_2(\mathcal L)$ dense. For a window $W\subset H$, define \[ \Lambda(W)=\{\pi_1(\ell):\ell\in\mathcal L,\ \pi_2(\ell)\in W\}. \] The usual regular-window convention takes $W$ compact with nonempty interior, $W=\overline{\operatorname{int}W}$, and Haar measure $m_H(\partial W)=0$. Translates may be included in the model-set convention. Regularity is an additional window condition: merely arising from a cut-and-project construction does not impose zero boundary measure. Singular boundary choices, generic windows, and inter-model-set conventions need specification.

The question is to give necessary-and-sufficient conditions expressed from $\Lambda$ itself, or its translation dynamical system, for the existence of such a regular cut-and-project realization, without presupposing that realization. For finite-local-complexity sets, the translation hull is the closure of $\{\Lambda-t:t\in{\mathbb{R}}^d\}$ in the usual local patch topology. Repetitivity means every finite patch recurs relatively densely. These are possible intrinsic data, not extra assumptions silently imposed on every version of the question.

The supplied literature triage dated 21 August 2026 reports several precise dynamical and almost-automorphic characterizations of regular and repetitive model sets. It does not supply a particular theorem with its hypotheses, so no equivalence involving almost automorphy, spectrum, or a maximal equicontinuous factor is asserted here without those hypotheses. The intended class may require repetitivity, finite local complexity, or other conditions depending on the chosen theorem. The word "simple" is not given a formal criterion. The source request and its partial-resolution assessment therefore remain broader than any uniquely specified characterization in the supplied record.

Short English statement

The source seeks criteria intrinsic to a Delone set or its translation hull that recognize a cut-and-project realization with a regular window. The supplied curation reports known dynamical characterizations, but does not provide their exact hypotheses or a formal meaning of simplicity.

Sources

• Local UnsolvedMath registry, permanent ID $30000016$, exact source code catalogue entry, Oberwolfach Reports Open Problems. Statement status exact; supplied verification finds no defensible textual repair necessary.
• Supplied report: "Mathematics and Physics of Disordered Systems (2005)," Oberwolfach Reports, pp. 1167--1232, DOI 10.4171/owr/2004/22: https://doi.org/10.4171/owr/2004/22. Proposed-year metadata and DOI refer to 2004, while the citation label says 2005.
• Supplied literature source "Baake--Lenz--Moody": https://arxiv.org/abs/math/0511648.
• Supplied "Lee--Moody": https://arxiv.org/abs/math/0510426.
• Supplied "Aujogue": https://www.cambridge.org/core/product/identifier/S0008439500023961/type/journal_article. Fuller bibliographic entries and theorem statements are not supplied.
• Literature checked 21 August 2026: supplied status partially_solved, with dynamical/almost-automorphic characterizations known in precise settings. Registry created 25 August 2026 and updated 16 September 2026. This dated curation triage requires independent verification before citation.
• Registry curation attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Underlying report and literature retain their supplied provenance; this statement-only draft requires human review.