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

Piyali Chakraborty, Dorin Ervin Dutkay

TL;DR

The paper investigates Plancherel-type dualities for unbounded subsets of $\mathbb{R}$ within Fuglede’s spectral conjecture framework, focusing on one-dimensional tilings by the finite set $\{0,1,\dots,p-1\}$. It proves that such tilings are equivalent to spectrality with a concrete dual measure, namely $\mu= p\,\mathrm{Leb}_{[-\frac{1}{2p},\frac{1}{2p}]+\mathbb{Z}}$, thereby establishing a precise Plancherel identity for these pairs. The proof structure decomposes $\Omega$ as $\Omega_0+p\mathbb{Z}$, constructs finite approximants with explicit spectra, and uses a standard spectral-lemma to pass to the limit, followed by a converse via periodization and disjointness arguments to deduce tiling from spectrality. These results introduce a new class of unbounded spectral sets in $\mathbb{R}$ and extend Fuglede-type dualities to infinite-measure, one-dimensional settings with an explicit dual lattice-like spectrum.

Abstract

In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of $\mathbb{R}$ that are in duality. In the terminology commonly used in the context of Fuglede's conjecture, our result states that an open set tiles $\mathbb{R}$ by the finite set $\{0,1,\dots,p-1\}$ if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.

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

TL;DR

The paper investigates Plancherel-type dualities for unbounded subsets of within Fuglede’s spectral conjecture framework, focusing on one-dimensional tilings by the finite set . It proves that such tilings are equivalent to spectrality with a concrete dual measure, namely , thereby establishing a precise Plancherel identity for these pairs. The proof structure decomposes as , constructs finite approximants with explicit spectra, and uses a standard spectral-lemma to pass to the limit, followed by a converse via periodization and disjointness arguments to deduce tiling from spectrality. These results introduce a new class of unbounded spectral sets in and extend Fuglede-type dualities to infinite-measure, one-dimensional settings with an explicit dual lattice-like spectrum.

Abstract

In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of that are in duality. In the terminology commonly used in the context of Fuglede's conjecture, our result states that an open set tiles by the finite set if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on .

Paper Structure

This paper contains 2 sections, 16 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm']}

Key Result

Theorem 1.1

Fug74 Let $\Omega\subset\mathbb{R}^d$ be an open, connected Nikodym region of finite measure. Define There exist commuting self-adjoint extensions of the differential operators $\{D_j: j=1,\dots,d\}$ if and only if there exists a subset $\Lambda$ of $\mathbb{R}^d$ such that the family of exponential functions forms an orthogonal basis for $L^2(\Omega)$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 25 more