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

TL;DR

This paper explores Plancherel identities for unbounded sets in real numbers related to Fuglede's conjecture, establishing conditions under which such sets tile the real line and possess spectra, with implications for Fourier analysis.

Contribution

It proves new Plancherel identities and demonstrates the surjectivity of the Fourier transform for specific unbounded tiling sets in duality, advancing understanding in harmonic analysis and tiling theory.

Findings

Open sets tile $ $ with finite sets iff they have a specific spectrum.

Fourier transform is surjective between certain unbounded tiling sets.

Results relate tiling properties to spectral measures in $ $.

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 $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.