Assume that \(\mu\) is a finite measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R})\) ). Show that

(i) \(\exists \xi eq 0: \widehat{\mu}(\xi)=\widehat{\mu}(0) \Longleftrightarrow \exists \xi eq 0: \mu(\mathbb{R} \backslash(2 \pi / \xi) \mathbb{Z})=0\);

(ii) \(\exists \xi_{1}, \xi_{2}, \xi_{1} / \xi_{2} otin \mathbb{Q}:\left|\widehat{\mu}\left(\xi_{1}ight)ight|=\left|\widehat{\mu}\left(\xi_{2}ight)ight|=\widehat{\mu}(0) \Longrightarrow|\widehat{\mu}| \equiv \widehat{\mu}(0)\).