Some Plancherel identities for unbounded subsets of $\mathbb R$ in duality

This paper establishes Plancherel-type identities and proves the surjectivity of the Fourier transform for certain unbounded dual tiling sets in $\mathbb{R}$, demonstrating that an open set tiles the real line by the finite set $\{0,1,\dots,p-1\}$ if and only if it admits a spectrum given by the Lebesgue measure on $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.

The Gist

Imagine you have a giant, infinite floor (the real number line, $\mathbb{R}$) and a collection of unique, oddly shaped tiles. The big question mathematicians have been asking for decades is: How do you know if a shape can perfectly cover this infinite floor without any gaps or overlaps?

This paper by Piyali Chakraborty and Dorin Ervin Dutkay solves a specific, tricky version of this puzzle. They look at shapes that are infinite (they stretch on forever) and prove a surprising rule about how they fit together.

Here is the story of their discovery, broken down into simple concepts.

1. The Two Ways to Look at a Shape

The authors connect two very different ways of describing a shape:

• The Tiling Way (The Floor): Can you slide copies of this shape around (using specific steps) to cover the entire floor perfectly?
  • Analogy: Imagine you have a weirdly shaped cookie cutter. If you press it down at specific spots (like 0, 1, 2, 3...), do the cookies cover the whole table without overlapping?
• The Spectrum Way (The Sound): Can you break this shape down into a pure "musical chord" made of specific frequencies?
  • Analogy: Imagine the shape is a drum. If you hit it, does it vibrate at a specific set of notes that are perfectly harmonious (orthogonal) and cover all possible sounds?

The Big Question: Does a shape that tiles the floor always have a perfect musical chord, and vice versa? This is known as Fuglede's Conjecture. It was proven true for small, finite shapes, but for infinite shapes, it was a mystery.

2. The Specific Puzzle They Solved

The authors focused on a very specific type of infinite shape in one dimension (a line).

• The Tiling Rule: They looked at shapes that tile the line using the steps $\{0, 1, 2, ..., p-1\}$.
  • Visual: Imagine your shape is a long, jagged strip. If you take that strip and slide it forward by 0 units, 1 unit, 2 units... up to $p-1$ units, these $p$ copies fit together perfectly to make a solid, unbroken line.
• The Result: They proved that if a shape does this tiling trick, it automatically has a perfect "musical chord" (a spectrum).

3. The "Magic Mirror" (The Duality)

The most beautiful part of their discovery is the duality. They found that the "tiling steps" and the "musical notes" are like reflections in a magic mirror.

• The Tiling Steps: You use the integers $0, 1, ..., p-1$.
• The Musical Notes (The Spectrum): The notes are located on a very specific, repeating pattern of tiny intervals:
  $$\left[ -\frac{1}{2p}, \frac{1}{2p} \right] + \mathbb{Z}$$
  • Visual: Imagine a ruler. You only care about the tiny slivers of the ruler that are right in the middle of every integer (from -0.5/p to 0.5/p, then -1.5/p to 1.5/p, etc.).
• The Connection: The size of your tiling steps ($p$) determines the size of these musical slivers. If you take $p$ steps to tile, your music lives in slivers that are $1/p$ the size of a standard unit.

4. How They Proved It (The "Zoom" Method)

Proving this for an infinite shape is hard because you can't just count the pieces. The authors used a clever "zoom" technique:

1. The Finite Slice: They took a huge chunk of their infinite shape and looked at it as if it were a finite piece.
2. The Approximation: They showed that as they made this chunk bigger and bigger (adding more and more copies of the shape), the "music" of the chunk started to look exactly like the "music" of the infinite shape.
3. The Limit: They proved that in the limit (when the chunk becomes infinite), the math works out perfectly. The "energy" of the shape (its size) matches the "energy" of the sound waves exactly. This is called a Plancherel Identity—a fancy way of saying "What you put in (the shape) equals what you get out (the sound)."

5. Why This Matters

Before this paper, we knew how to tile finite shapes and how to tile the whole infinite line. But we didn't know how to tile "in-between" infinite shapes that repeat in a specific pattern.

This paper says: "If you can tile the line with a specific set of steps, you automatically have a perfect mathematical harmony."

It's like discovering that if you can arrange a set of bricks to build a wall without gaps, the wall will naturally hum a perfect, resonant tone. It connects the geometry of space (tiling) with the physics of waves (spectra) in a way that works even for shapes that go on forever.

Summary in One Sentence

The authors proved that for a specific type of infinite shape, the ability to tile a line with a set of integer steps is mathematically identical to the ability to generate a perfect, repeating musical spectrum, revealing a deep symmetry between how shapes fit together and how they vibrate.

Technical Summary

Here is a detailed technical summary of the paper "Some Plancherel Identities for Unbounded Subsets of $\mathbb{R}$ in Duality" by Piyali Chakraborty and Dorin Ervin Dutkay.

1. Problem Statement and Context

The paper addresses a specific case of Fuglede's Conjecture, which posits that a measurable set $\Omega \subset \mathbb{R}^d$ of finite measure is spectral (admits an orthogonal basis of exponential functions in $L^2(\Omega)$) if and only if it tiles $\mathbb{R}^d$ by translations.

While the conjecture is known to be false for general sets in dimensions $d \geq 3$, it remains an active area of research for specific classes of sets and dimensions. Previous work by Fuglede (1974) and Pedersen (1987) established the equivalence for connected domains, with Pedersen extending the definition of spectral sets to include infinite-measure domains via spectral pairs $(\Omega, \mu)$.

The Specific Gap:
Most existing results for infinite-measure sets involve lattice structures or specific sublattices. This paper investigates a novel class of unbounded spectral sets in $\mathbb{R}$ that are not simple lattices but are constructed via tiling with a finite set. Specifically, the authors aim to characterize open sets $\Omega \subset \mathbb{R}$ that tile the real line using the finite translation set $\{0, 1, \dots, p-1\}$ (where $p \in \mathbb{N}, p \geq 2$) and determine their corresponding spectral properties.

2. Methodology

The authors employ a combination of harmonic analysis, measure theory, and operator theory. The core methodology involves:

• Spectral Pair Definition: Utilizing Pedersen's definition where $(\Omega, \mu)$ is a spectral pair if the Fourier transform $f \mapsto \hat{f}$ extends to an isometric isomorphism from $L^2(\Omega)$ to $L^2(\mu)$.
• Approximation by Finite Sets: To handle the unbounded nature of $\Omega$, the authors construct an increasing sequence of bounded subsets $\Omega_n \subset \Omega$ that cover $\Omega$. They analyze the spectral properties of these finite approximations.
• Periodization and Convolution: The proof relies heavily on the properties of periodization ($Per(f)$) and the convolution of measures. The authors relate the tiling property to the periodicity of the Fourier transform of the characteristic function of the set.
• Plancherel-Type Identities: The central technical tool is establishing a specific Plancherel identity that links the $L^2$ norm on the spatial domain $\Omega$ to an $L^2$ norm on a specific frequency domain measure.
• Density Arguments: The proof of surjectivity (onto-ness) of the Fourier transform uses density arguments involving Schwartz functions and inverse Fourier transforms to show that the image of the transform covers the target space.

3. Key Contributions and Results

Main Theorem (Theorem 1.8)

The primary result establishes a precise duality between tiling and spectrality for unbounded sets in $\mathbb{R}$:

Let $\Omega$ be an open subset of $\mathbb{R}$ and $p \in \mathbb{N}, p \geq 2$. Then $\Omega$ tiles $\mathbb{R}$ by the finite set $\{0, 1, \dots, p-1\}$ if and only if $\Omega$ is a spectral set with the pair measure $\mu$ given by:
$$\mu = p \cdot \text{Leb}_{\left[-\frac{1}{2p}, \frac{1}{2p}\right] + \mathbb{Z}}$$
(i.e., $p$ times the Lebesgue measure restricted to the union of intervals centered at integers with radius $1/2p$).

Technical Breakdown of the Proof

The proof is divided into two directions:

A. Direct Implication (Tiling $\implies$ Spectral):

1. Decomposition: If $\Omega$ tiles $\mathbb{R}$ by $\{0, \dots, p-1\}$, the authors define $\Omega_0 = \Omega \cap [0, p)$. They prove that $\Omega_0$ tiles $\mathbb{R}$ by $\mathbb{Z}$ and is congruent to $[0, 1)$ modulo $\mathbb{Z}$.
2. Approximation: They construct $\Omega_n = \Omega_0 + p\{-n, \dots, n\}$. Using Lemma 2.7 (a criterion for spectral measures involving the sum of squared Fourier transforms), they show that $\Omega_n$ has a spectrum $\Lambda_n = \frac{1}{p(2n+1)}\{-n, \dots, n\} + \mathbb{Z}$.
3. Isometry: By taking the limit as $n \to \infty$, they demonstrate that the Fourier transform maps $L^2(\Omega)$ isometrically into $L^2(\mu)$, where $\mu$ is the renormalized Lebesgue measure on $\left[-\frac{1}{2p}, \frac{1}{2p}\right] + \mathbb{Z}$.
4. Surjectivity: They prove the map is onto by constructing pre-images for functions supported on the spectral measure using the periodicity of the Fourier transform of the characteristic function of $\Omega$.

B. Converse Implication (Spectral $\implies$ Tiling):

1. Periodicity: Assuming $\Omega$ is spectral with the specific measure $\mu$, they prove that the periodization of the product of Fourier transforms, $Per(\hat{f}\hat{g})$, has a period of $1/p$.
2. Disjointness: Using this periodicity and Plancherel's identity, they show that $\Omega \cap (\Omega + j)$ has measure zero for any $j \not\equiv 0 \pmod p$.
3. Construction of a Tiling Set: They define a superset $\tilde{\Omega}$ constructed from $\Omega$ that tiles $\mathbb{R}$ by $\{0, \dots, p-1\}$.
4. Equality: Using a contradiction argument based on the surjectivity of the Fourier transform, they prove that the complement $\tilde{\Omega} \setminus \Omega$ must have measure zero, implying $\Omega = \tilde{\Omega}$ almost everywhere. Thus, $\Omega$ itself tiles $\mathbb{R}$.

4. Significance

• New Class of Examples: The paper provides a new, explicit class of unbounded spectral sets in $\mathbb{R}$ that are not simple lattices. This expands the known landscape of spectral sets beyond the finite-measure and lattice cases.
• Refinement of Fuglede's Conjecture: While Fuglede's conjecture is false in general, this work confirms its validity for the specific, non-trivial case of sets tiling by finite sets in one dimension. It clarifies the exact structure of the "dual" measure required for such sets.
• Operator Theory Connection: The results reinforce the link between the existence of commuting self-adjoint extensions of differential operators (Segal's original question) and the geometric tiling properties of the domain, even in the unbounded setting.
• Methodological Advancement: The technique of approximating unbounded spectral sets with bounded, lattice-based spectral sets and taking limits to derive Plancherel identities offers a robust framework for analyzing other unbounded domains in harmonic analysis.

In summary, the paper successfully establishes a rigorous duality between tiling by finite sets and spectrality with a specific periodic measure for unbounded subsets of the real line, contributing significantly to the understanding of the spectral-tiling relationship in infinite measure contexts.