The spectrality of symmetric additive measures

Ai, Wen-Hui; Lu, Zheng-Yi; Zhou, Ting

Comptes Rendus. Mathématique, Tome 361 (2023) no. G4, pp. 783-793

Let $\rho$ be a symmetric measure of Lebesgue type, i.e.,

$$\rho =\frac{1}{2}(\mu \times {\delta }_0+{\delta }_0\times \mu ),$$

where the component measure $\mu$ is the Lebesgue measure supported on $[t,t+1]$ for $t\in \mathbb{Q}\setminus \{-\frac{1}{2}\}$ and ${\delta }_0$ is the Dirac measure at $0$. We prove that $\rho$ is a spectral measure if and only if $t\in \frac{1}{2}\mathbb{Z}$. In this case, $L^2\left(\rho \right)$ has a unique orthonormal basis of the form

$$\left\{e^{2\pi i(\lambda x-\lambda y)}:\lambda \in {\Lambda }_0\right\},$$

where ${\Lambda }_0$ is the spectrum of the Lebesgue measure supported on $[-t-1,-t]\cup [t,t+1]$. Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021].

Classification : 28A80, 42C05

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