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

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 $[-\frac{1}{2p},\frac{1}{2p}]+\mathbb{Z}$.