Cut-and-Project Systems Explained

• Cut-and-project systems are geometric constructions that generate aperiodic model sets by projecting lattice points selected through an internal space window.
• They integrate Euclidean, LCAG, and homogeneous dynamics approaches to analyze structural properties, diffraction, and substitutional organization.
• Applications span quasicrystal modeling, harmonic analysis, and non-Euclidean variants, highlighting insights into discrepancy theory and self-similarity.

Searching arXiv for recent and foundational papers on cut-and-project systems. Cut-and-project systems are geometric constructions that produce structured point sets in lower-dimensional “physical” spaces from lattices in higher-dimensional spaces by selecting lattice points through a window in an “internal” space and projecting them. In the Euclidean setting, they arise from a lattice $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ or, more generally, from a cut and project scheme $(G,H,\mathcal{L})$ in locally compact abelian groups, with injective physical projection and dense internal projection. The resulting sets—often called model sets—form a central class in aperiodic order, mathematical quasicrystals, discrepancy theory, and homogeneous dynamics, and they also appear in recent work on Fourier transformable measures, substitutional structure, self-similarity, and even quantum models of quasicrystals (Etkind et al., 26 Nov 2025, Strungaru, 2021, Rühr et al., 2020, Nop et al., 15 Mar 2026).

1. Euclidean and LCAG formulations

A Euclidean cut-and-project system begins with positive integers $m,n$, the product space

$\mathbb{R}^m \times \mathbb{R}^n,$

and a lattice $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$. The coordinate projections are

$$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$

In the standard cut-and-project setting, $p_1|_\Gamma$ is injective and $p_2(\Gamma)$ is dense in $\mathbb{R}^n$. This is the typical “general position” assumption: the lattice is skew relative to the coordinate splitting so that neither projection collapses too much structure (Etkind et al., 26 Nov 2025).

An equivalent formulation, used in the LCAG setting, defines a cut and project scheme as a triple $(G,H,\mathcal{L})$ consisting of a $(G,H,\mathcal{L})$0-compact LCAG $(G,H,\mathcal{L})$1, a LCAG $(G,H,\mathcal{L})$2, and a lattice $(G,H,\mathcal{L})$3 such that $(G,H,\mathcal{L})$4 is dense in $(G,H,\mathcal{L})$5 and the restriction $(G,H,\mathcal{L})$6 is one to one. Writing

$(G,H,\mathcal{L})$7

the first projection induces a group isomorphism between $(G,H,\mathcal{L})$8 and $(G,H,\mathcal{L})$9, and hence a $m,n$0-mapping

$m,n$1

with

$m,n$2

This formulation is particularly useful for harmonic analysis and Meyer sets, because it identifies lattice points with their physical and internal coordinates simultaneously (Strungaru, 2021).

A third formulation, important in homogeneous dynamics, fixes integers $m,n$3, $m,n$4, sets

$m,n$5

and uses the projections $m,n$6 and $m,n$7. In this language, irreducibility consists of density of the internal projection, injectivity of the physical projection, and regularity of the window. This perspective places cut-and-project sets in the space $m,n$8 of closed subsets endowed with the Chabauty–Fell topology, where they can be studied statistically under $m,n$9 and $\mathbb{R}^m \times \mathbb{R}^n,$0 actions (Rühr et al., 2020).

These formulations are compatible rather than competing. This suggests that “cut-and-project system” is best understood as a structural mechanism rather than a single rigid definition: Euclidean model sets, LCAG model sets, and homogeneous-dynamical parameter spaces are different realizations of the same underlying geometry.

2. Model sets, windows, and regularity

Given a cut-and-project scheme and a bounded set $\mathbb{R}^m \times \mathbb{R}^n,$1 in internal space, the associated point set in physical space is defined by

$\mathbb{R}^m \times \mathbb{R}^n,$2

or, in the LCAG notation,

$\mathbb{R}^m \times \mathbb{R}^n,$3

This is the central formula of the theory: one cuts the lattice by the strip determined by the window and projects the selected points to physical space (Etkind et al., 26 Nov 2025, Strungaru, 2021).

Window regularity controls the geometry and dynamics of the resulting model set. In the Euclidean literature, the paper on bounded distance equivalence assumes windows $\mathbb{R}^m \times \mathbb{R}^n,$4 are bounded, Riemann measurable, and of positive measure, where Riemann measurable means that the boundary has Lebesgue measure zero. In the LCAG setting, compact windows define weak model sets, and compact windows with non-empty interior define model sets (Etkind et al., 26 Nov 2025, Strungaru, 2021).

Under standard assumptions, these sets are uniformly discrete and relatively dense, hence Delone. In the classical picture of aperiodic order, one writes

$\mathbb{R}^m \times \mathbb{R}^n,$5

with physical space $\mathbb{R}^m \times \mathbb{R}^n,$6, internal space $\mathbb{R}^m \times \mathbb{R}^n,$7, and total space $\mathbb{R}^m \times \mathbb{R}^n,$8. For suitable $\mathbb{R}^m \times \mathbb{R}^n,$9, the resulting $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$0 is a Delone set, the associated dynamical system is minimal and uniquely ergodic, and the diffraction is pure point, with Bragg peaks supported on the dual of the lattice (Etkind et al., 26 Nov 2025).

The theory also accommodates windows that are not smooth or polytopal. Recent constructions use self-similar attractors as windows, obtained from a contracting conjugate iterated function system. In that setting, if $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$1 is the attractor of the conjugate system, then the model set

$\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$2

is the maximal self-similar solution of a physical-space inflation equation. This extends the class of acceptance domains from intervals, polygons, and polyhedra to heavily overlapping self-similar fractals (Bandt et al., 24 May 2026).

By contrast, some recent work shows that windows with thick boundary can produce Euclidean model sets with positive topological entropy, including both proper windows and windows with empty interior. A plausible implication is that regularity of the boundary is not merely technical: it can separate pure low-complexity order from model sets that still arise from a cut-and-project scheme but display substantial combinatorial complexity (Jäger et al., 2016).

3. Structural classes: Meyer sets, substitutional systems, and self-similarity

Meyer sets are one of the principal structural classes attached to cut-and-project systems. A set $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$3 is a Meyer set if $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$4 is relatively dense and $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$5 is uniformly discrete. A key characterization states that a relatively dense set is Meyer if and only if it is a subset of a weak model set. Thus cut-and-project systems provide the natural framework for studying Meyer sets and measures supported on them (Strungaru, 2021).

Substitutional structure is another major theme. A recent characterization in Euclidean total space answers when a cut-and-project set may also be defined by a substitution rule. The result states that the cut-and-project sets in a hull are $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$6-Sub if and only if there exists a linear map $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$7 on total space such that $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$8, $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$9 with $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$0, $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$1, and a translate $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$2 of the window is mutually constructable from $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$3, where $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$4 (Harriss et al., 15 Dec 2025).

For polytopal windows this criterion simplifies: after passing to a power $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$5, substitutionality is equivalent to invariance of supporting hyperplane directions under the internal contraction and rationality of the window. In codimension one, a 2-to-1 cut-and-project scheme with $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$6 is substitutional if and only if the slope is quadratic irrational. This places classical Sturmian and substitutional one-dimensional quasicrystals inside a common framework (Harriss et al., 15 Dec 2025).

A related but distinct question concerns generalized self-similarities. For a cut-and-project set $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$7, a linear map $$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$8 is a self-similarity if

$$p_1 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^m, \qquad p_2 : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n.$$9

The paper on generalized self-similarities characterizes when such an $p_1|_\Gamma$0 can occur. If $p_1|_\Gamma$1 is a self-similarity of a generic scheme, then there exist matrices $p_1|_\Gamma$2 and $p_1|_\Gamma$3 with $p_1|_\Gamma$4 such that

$p_1|_\Gamma$5

In particular, every eigenvalue of $p_1|_\Gamma$6 is an algebraic integer. For $p_1|_\Gamma$7 diagonalizable over $p_1|_\Gamma$8, existence of a generic cut-and-project scheme with self-similarity $p_1|_\Gamma$9 is equivalent to the requirement that the spectrum of $p_2(\Gamma)$0 consist entirely of algebraic integers (Masáková et al., 2019).

These results show that substitutional and self-similar cut-and-project systems are controlled simultaneously by geometry of the window and arithmetic of the linear action on the lattice. This suggests that the “substitutional” subclass of model sets is rigid both combinatorially and arithmetically.

4. Quantitative structure: bounded distance, discrepancy, and linear repetitivity

Two cut-and-project sets $p_2(\Gamma)$1 are bounded distance equivalent if there exists a bijection

$p_2(\Gamma)$2

such that

$p_2(\Gamma)$3

This is a strong form of equivalence: every point can be moved by a uniformly bounded amount. Recent work proves that if two model sets from the same lattice and Riemann measurable windows are bounded distance equivalent, then the windows are equidecomposable up to measure zero with measurable pieces, using only translations from $p_2(\Gamma)$4 (Etkind et al., 26 Nov 2025).

In one-dimensional physical space there is a stronger conclusion: if $p_2(\Gamma)$5 and $p_2(\Gamma)$6 are bounded distance equivalent, then the windows are equidecomposable up to measure zero with Riemann measurable pieces, and in the polytope case even with polytope pieces, by translations from $p_2(\Gamma)$7. This is achieved by a different method linked to bounded remainder sets (Etkind et al., 26 Nov 2025).

The connection with discrepancy theory is classical and explicit. For an irrational vector $p_2(\Gamma)$8, a bounded measurable set $p_2(\Gamma)$9 is a bounded remainder set if the discrepancy

$\mathbb{R}^n$0

is uniformly bounded in $\mathbb{R}^n$1 for almost every $\mathbb{R}^n$2. For one-dimensional model sets, a linear image of the window is a bounded remainder set precisely when the model set is bounded distance equivalent to an arithmetic progression (Etkind et al., 26 Nov 2025).

This perspective underlies the construction of windows whose associated cut-and-project sets are bounded distance to lattices. For any totally irrational cut-and-project setup, one can construct an infinite collection of windows using special regions for toral rotations, and the associated cut-and-project sets are bounded distance to lattices (Haynes et al., 2014). In codimension one, it is possible to choose sections in non-trivial ways so that the resulting sets are bounded displacement to lattices, and the corresponding discrepancy for $\mathbb{R}^n$3 along certain special intervals is uniformly bounded by a constant, regardless of Diophantine type (Haynes, 2013).

Linear repetitivity provides a more dynamical quantitative invariant. For cut-and-project sets with convex polytopal windows, linear repetitivity is completely classified within a broad weakly homogeneous class: it holds if and only if the system has low complexity (property C) and a Diophantine condition (property D) on the projected lattice (Koivusalo et al., 2020). For typical cubical cut-and-project sets, one has explicit discrepancy estimates for patch frequencies and quantitative bounds for repetitivity and repulsivity, obtained by discrepancy-theoretic methods (Haynes et al., 2017).

The quantitative theory therefore links three notions that often appear separately: bounded distance equivalence, bounded remainder discrepancy, and repetitivity. A plausible synthesis is that all three measure, in different languages, how close a model set is to periodic or substitutional organization without becoming periodic.

5. Harmonic analysis, diffraction, and statistical descriptions

Cut-and-project systems are fundamental in diffraction theory because Fourier analysis on the higher-dimensional lattice can be transferred to measures on the physical-space model set. For a CPS $\mathbb{R}^n$4 with compact window $\mathbb{R}^n$5, the paper on Meyer-set-supported measures defines two spaces: $\mathbb{R}^n$6 the latter consisting of translation-bounded measures supported on the strip $\mathbb{R}^n$7. The lift and descent operators give a bijection between them (Strungaru, 2021).

The main theorem there states that a translation-bounded measure $\mathbb{R}^n$8 supported inside $\mathbb{R}^n$9 is Fourier transformable if and only if its lift $(G,H,\mathcal{L})$0 to the strip is Fourier transformable. Moreover, if $(G,H,\mathcal{L})$1 equals $(G,H,\mathcal{L})$2 on $(G,H,\mathcal{L})$3, then for all $(G,H,\mathcal{L})$4,

$(G,H,\mathcal{L})$5

This expresses the Fourier transform of a measure on a model set in terms of the Fourier transform of a measure on the higher-dimensional lattice strip (Strungaru, 2021).

The dual cut-and-project scheme then organizes diffraction in frequency space. In that setting, a Fourier transformable measure with Meyer set support can be represented as

$(G,H,\mathcal{L})$6

where $(G,H,\mathcal{L})$7 is a periodic measure on the dual lattice and $(G,H,\mathcal{L})$8 is a strongly admissible internal-space function. This leads to norm almost periodicity of $(G,H,\mathcal{L})$9 and to a generalized Eberlein decomposition compatible with the model-set structure (Strungaru, 2021).

A different statistical viewpoint treats cut-and-project sets themselves as random variables under invariant measures. Ratner–Marklof–Strömbergsson measures are Borel probability measures on the space of closed subsets of $(G,H,\mathcal{L})$00 that are supported on irreducible cut-and-project sets and invariant and ergodic under $(G,H,\mathcal{L})$01 or $(G,H,\mathcal{L})$02. These measures are classified via homogeneous dynamics and algebraic groups, and they satisfy analogues of Siegel, Weil, and Rogers formulas (Rühr et al., 2020).

Within that framework, typical cut-and-project sets satisfy point-counting and patch-counting asymptotics with explicit error estimates. For higher-rank RMS measures satisfying the stated condition, for almost every cut-and-project set $(G,H,\mathcal{L})$03,

$(G,H,\mathcal{L})$04

and analogous patch-counting estimates hold under a box-dimension assumption on the window boundary (Rühr et al., 2020).

At the opposite end of the regularity spectrum, Euclidean model sets with thick-boundary windows can have positive topological entropy. The constructions with proper random windows and with windows of empty interior show that the entropy can be proportional to the measure of the boundary of the window (Jäger et al., 2016). This provides an objective correction to a common simplification: cut-and-project sets are not invariably zero-entropy or pure low-complexity objects.

6. Extensions, applications, and non-Euclidean variants

Cut-and-project systems are central in aperiodic order, but recent work pushes them into neighboring fields. One algebraic extension constructs Jordan algebras over icosahedral quasicrystals obtainable as model sets of a cut-and-project scheme with a convex acceptance window. In that framework, the lattice is based on the icosian ring or the $(G,H,\mathcal{L})$05 lattice, the model set is

$(G,H,\mathcal{L})$06

and convexity of $(G,H,\mathcal{L})$07 implies closure under a quasiaddition operation, which in turn defines a Jordan product on generators indexed by quasicrystal points (Corradetti et al., 2023).

This approach preserves non-crystallographic symmetries $(G,H,\mathcal{L})$08, $(G,H,\mathcal{L})$09, and $(G,H,\mathcal{L})$10: if the acceptance window enjoys the corresponding symmetry, then the resulting Jordan algebra enjoys the same symmetry. The paper works out explicit examples for the Fibonacci chain, Penrose tiling, a $(G,H,\mathcal{L})$11-quasicrystal, and the Elser–Sloane quasicrystal (Corradetti et al., 2023).

A physical extension is provided by cut-and-project density functional theory for quasicrystals. There the standard higher-dimensional cut-and-project geometry is made operational for interacting quantum systems by embedding physical space as a subspace $(G,H,\mathcal{L})$12, introducing lifting and lowering maps between physical and higher space, and replacing derivatives by $(G,H,\mathcal{L})$13 in the higher-dimensional periodic problem. The resulting DFT++ formulation treats quasicrystals as crystals in higher space and avoids crystalline approximants (Nop et al., 15 Mar 2026).

The formalism is illustrated for the Fibonacci quasicrystal, where higher space is $(G,H,\mathcal{L})$14, physical space is $(G,H,\mathcal{L})$15, the lattice is $(G,H,\mathcal{L})$16, and the cut-and-project geometry is encoded by atomic segments orthogonal to the embedded line. This permits direct specification of quasicrystalline quantum states and density of states from higher-space band data (Nop et al., 15 Mar 2026).

A more radical extension replaces Euclidean ambient geometry by hyperbolic geometry. In the Poincaré disc, a cocompact Fuchsian group orbit plays the role of the lattice, a geodesic and its tubular neighborhood play the role of the slicing data, and the projected set on the geodesic gives a one-dimensional point set $(G,H,\mathcal{L})$17. Under explicit conditions on the fundamental domain, the resulting set is a chaotic Delone set, and its tile-length set is countably infinite (Howat et al., 11 Mar 2026).

Finally, self-similar attractors can be used directly as windows. Starting from a Pisot unit $(G,H,\mathcal{L})$18 and an iterated function system

$(G,H,\mathcal{L})$19

the attractor of the conjugate contracting system becomes the acceptance window, and the corresponding cut-and-project model set in $(G,H,\mathcal{L})$20 satisfies

$(G,H,\mathcal{L})$21

This approach emphasizes overlaps as a source of natural decoration rather than as an obstruction (Bandt et al., 24 May 2026).

These developments suggest that the notion of cut-and-project system has become a platform rather than a niche construction: it now supports classification theorems, discrepancy theory, harmonic analysis, algebraic structures, statistical mechanics, and non-Euclidean generalizations, while retaining the same core mechanism of cutting in internal space and projecting to physical space.