Spectral Sets: Theory & Applications

• Spectral sets are subsets of Euclidean space whose L²-space admits an orthogonal basis of exponential functions, serving as a fundamental definition in harmonic analysis.
• They are deeply connected to the Fuglede conjecture and tiling theory, with convex bodies yielding proper tiling and general sets exhibiting weak tiling properties.
• Recent studies extend spectral set theory to operator theory and optimization, highlighting periodicity in unions of intervals and applications in quantum information.

A spectral set is a subset of Euclidean space (or a more general mathematical structure) with the property that its $L^2$-space admits an orthogonal basis of exponential functions. This notion connects harmonic analysis, operator theory, convex geometry, combinatorics, functional analysis, and quantum theory. Spectral sets appear in the study of tilings, operator dilations, self-similar structures, semidefinite optimization, and noncommutative geometry, among other areas. The theory has deep links to the celebrated Fuglede conjecture, to spectral synthesis in Abstract Harmonic Analysis, and to information-theoretic properties of convex state spaces.

1. Classical Definition and Fundamental Properties

Let $\Omega\subset\mathbb{R}^d$ be a bounded measurable set of positive measure. $\Omega$ is called a spectral set if there exists a countable set $\Lambda\subset\mathbb{R}^d$ such that the family of functions

$$E(\Lambda) = \{e^{2\pi i \langle \lambda, x\rangle} : \lambda\in\Lambda\}$$

forms an orthonormal basis for $L^2(\Omega)$. This is equivalent to the two conditions:

• (Orthogonality) For $\lambda\ne\mu\in\Lambda$, $$\int_\Omega e^{2\pi i \langle\lambda-\mu,x\rangle}\,dx = 0$$,
• (Completeness) The closed linear span of $E(\Lambda)$ equals $L^2(\Omega)$.

The set $\Lambda$ is called a spectrum for $\Omega$ (Kolountzakis et al., 2022, Dutkay et al., 2012, Ducasse et al., 23 Jun 2025, Fu et al., 2013).

The notion generalizes to topological groups and even to discrete finite settings, such as finite abelian groups and finite fields (Aten et al., 2015). In matrix/operator theory, an analogous concept is a "spectral set" for an operator $T$ defined via norm inequalities for rational functions of $T$ (Badea et al., 2013).

2. Spectral Sets, Tiling, and the Fuglede Conjecture

The Fuglede conjecture (1974) posited that a set $\Omega$ is spectral if and only if it tiles the space by translations, i.e., there exists a (multi)set $T\subset\mathbb{R}^d$ such that almost every $x\in\mathbb{R}^d$ is covered exactly once by the family $\{\Omega+t: t\in T\}$. This conjecture holds in one dimension and for convex bodies in any dimension (Kolountzakis et al., 2022), but fails in higher dimensions for general sets (counterexamples in $d\geq3$) (Aten et al., 2015, Fu et al., 2013).

A central insight is that spectral sets always "weakly tile" their complement: there exists a positive measure $\nu$, not necessarily discrete, such that $1_\Omega*\nu=1_{\mathbb{R}^d\setminus \Omega}$ a.e. (Kolountzakis et al., 2022). For convex bodies, a spectral set is necessarily a proper translational tile, and the converse holds (Kolountzakis et al., 2022).

SPECIFIC HIERARCHY: | Class | Spectral $\Longrightarrow$ | Reference | |-----------------|---------------------------|---------------------| | Convex body | proper tiling | (Kolountzakis et al., 2022) | | Arbitrary set | weak tiling | (Kolountzakis et al., 2022) | | Cantor type | non-spectral (if weak tiling fails) | (Kolountzakis et al., 2022) |

Generalizations to product sets and self-similar tiles reveal nuanced behavior: products of spectral sets are spectral, but the converse can fail unless additional assumptions (e.g., near-interval structure) are imposed (Ramabadran et al., 21 Aug 2025). There is a full theory relating spectrality and tiling for cylindric and Cartesian product domains (Greenfeld et al., 2016, Kolountzakis et al., 2022).

3. Structural and Spectral Character Theorems

Periodicity and Structure in One Dimension

For unions of $n$ intervals in $\mathbb{R}$, any spectrum $\Lambda$ is periodic with period an integral multiple of $|\Omega|$. This rigidity gives rise to structure theorems: up to a scaling and translation, any such spectral set is a finite union of equal-length intervals aligned periodically (Bose et al., 2010). In the discrete case (i.e., on $\mathbb{Z}_n$ or $\mathbb{F}_p^d$), similar periodicity and structure hold, relating spectrality to cyclotomic and Hadamard-matrix structure (Aten et al., 2015).

Operator-Theoretic Notion: Spectral Sets for Operators

Let $A$ be a bounded linear operator on a Hilbert space, and $\Omega\subset\mathbb{C}$ a closed set containing $\sigma(A)$. $\Omega$ is a $K$-spectral set for $A$ if for every rational function $f$ bounded on $\Omega$,

$\|f(A)\| \leq K \max_{z\in\Omega}|f(z)|$

(Badea et al., 2013, Greenbaum et al., 2023, Crouzeix et al., 2018). When $K=1$, this recovers von Neumann’s spectral set definition (and, for the unit disk, the classical von Neumann inequality). The spectrum, resolvent, and numerical range interplay in controlling $K$ (Crouzeix et al., 2018, Greenbaum et al., 2023). For the numerical range, it is known that $K\leq 1+\sqrt{2}$ and conjectured that $K=2$ uniformly (Badea et al., 2013).

Dilation theorems characterize spectral sets in terms of the existence of normal (or isometric) dilations (Klaja et al., 2017, Jana et al., 2024). In Banach spaces, the equivalence between norm bounds, spectral sets, completions, and dilations breaks down in sharp ways, tightly linked to Bohr's inequality and fundamental Banach space properties (Jana et al., 2024).

Spectral Sets in Jordan Algebras and Quantum Information

A spectral set in a real Euclidean Jordan algebra $V$ is the inverse image under the eigenvalue map of a permutation-invariant subset of $\mathbb{R}^n$ (Gowda et al., 2018). Such sets have lifting properties: connectedness and irreducibility properties lift from the eigenvalue image to the Jordan algebra. In the context of convex state spaces, spectrality becomes equivalent to uniqueness of mixture coefficients in orthogonal decompositions—a requirement for the foundational principles of quantum information theory (Harremoës, 2017). In particular, only spectral sets support reversible measurements and information divergence with the expected properties.

4. Examples and Explicit Constructions

• Unions of intervals in $\mathbb{R}$: Complete structure and periodicity results (Bose et al., 2010, Dutkay et al., 2012, Ducasse et al., 23 Jun 2025).
• Self-similar and product-form tiles: For digit sets with a product structure, strict and modulo product-form tiles are always spectral, and the converse (spectral $\Longrightarrow$ tile) holds in low-cardinality settings (Fu et al., 2013).
• Cylindric domains: Spectrality passes between a cylindric set and its base (Greenfeld et al., 2016).
• Finite abelian and finite fields: In $\mathbb{F}_p^d$, for $d>2$, spectral sets may fail to tile and vice versa, with the obstruction given by the existence of log-Hadamard matrices of small rank (Aten et al., 2015).
• Convex hulls of spectral sets: Explicit projection-based convexification is available for general spectral constraints induced by spectral maps (e.g., eigenvalue and singular-value maps) and can be fully characterized via convex duality (Zhao, 2024).

5. Spectral Sets for Operators and Numerical Range

The classical spectral set theory for operators is broadened by considering numerical range analogues. For a Hilbert space operator $T$, a compact set $S$ is a spectral set if for every rational $f$ with poles off $S$, $\|f(T)\|\leq\sup_{z\in S}|f(z)|$. The numerical range $W(T)$ is a $K$-spectral set with $K=1+\sqrt{2}$, with connections to Crouzeix's theorem (Klaja et al., 2017, Crouzeix et al., 2018, Greenbaum et al., 2023). Key properties such as positivity criteria and operator dilations translate to this framework, but essential distinctions emerge:

• For the numerical-range-based spectral set, a base point in $S$ must be singled out,
• The operator norm is replaced by the numerical radius,
• The dilation gives $f(T)$ as a “2-dilation”—reflecting the bound $\|T\|\leq 2w(T)$ (Klaja et al., 2017).

Application to iterative methods (e.g., GMRES, Arnoldi) proceeds by employing $K$-spectral set bounds on the regions containing the spectrum and controlling convergence via rational approximation on $W(T)$ or general $K$-spectral sets (Crouzeix et al., 2018, Greenbaum et al., 2023, Badea et al., 2013).

6. Spectral Synthesis, Local Spectral Sets, and Abstract Harmonic Analysis

In the Fourier algebra $A(G)$ of a locally compact group $G$, a closed set $E\subset G$ is called a spectral set (or a set of spectral synthesis) if the ideal of functions vanishing on $E$ coincides with the ideal generated by compactly supported functions vanishing near $E$ (Ludwig et al., 2024). Local spectral sets are defined via approximation in $A(G)$ by functions vanishing near $E$. Ludwig and Turowska give a new Hilbert-space-based characterization of local spectral sets in terms of canonical $L^2$-factors and geometric squeezing conditions. For abelian $G$, the criterion is especially transparent, amounting to the ability to “move mass away” from $E$ in the $L^2$ sense (Ludwig et al., 2024).

Spectral synthesis is fundamental to understanding whether abstract sets can be reconstructed (synthesized) from exponential or trigonometric functions, and connects directly to the combinatorial and operator-theoretic properties of spectral sets in Euclidean settings—underlying, for example, finite unions and Ditkin set properties.

7. Convexification and Spectral Sets in Optimization

For spectral constraints $\lambda(x)\in C$ imposed by spectral maps $\lambda$ (eigenvalue, singular-value, hyperbolic polynomials), convexification of spectral sets is crucial in optimization. The closed convex hull of the set $S=\lambda^{-1}(C)$, for general $C$, can be exactly characterized by projection-based duality constructions (Zhao, 2024). In the presence of invariance (e.g., permutation or sign symmetry of $C$), convex hulls can be described even more simply. These results unify and extend classical theorems (e.g., Schur-Horn, von Neumann inequalities) and provide tractable convex relaxations for semidefinite and spectral optimization.

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References:

• (Bose et al., 2010) Spectrum is periodic for n-Intervals
• (Dutkay et al., 2012) Unitary groups and spectral sets
• (Badea et al., 2013) Spectral Sets (survey)
• (Dutkay et al., 2013) Local translations associated to spectral sets
• (Fu et al., 2013) Spectrality of Self-Similar Tiles
• (Aten et al., 2015) Tiling sets and spectral sets over finite fields
• (Greenfeld et al., 2016) Spectrality and tiling by cylindric domains
• (Klaja et al., 2017) Spectral sets for numerical range
• (Harremoës, 2017) Quantum Information on Spectral Sets
• (Crouzeix et al., 2018) Spectral Sets: Numerical Range and Beyond
• (Gowda et al., 2018) On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras
• (Kolountzakis et al., 2022) Spectral sets and weak tiling
• (Greenbaum et al., 2023) K-Spectral Sets
• (Zhao, 2024) On the Projection-Based Convexification of Some Spectral Sets
• (Jana et al., 2024) Spectral set, complete spectral set and dilation for Banach space operators
• (Ludwig et al., 2024) On the structure of Spectral Sets
• (Ducasse et al., 23 Jun 2025) Spectral properties of unions of intervals and groups of local translations
• (Ramabadran et al., 21 Aug 2025) Spectrality of Product Sets with a Perturbed Interval Factor