Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\displaystyle\int \mathrm{e}^{2i\pi t\cdot x} ~\mu(dx).$$

It is known that we don't necessarily have $\hat{\mu}(t)\rightarrow0$ for singular measure. But does one have like Lebesgue measure $$\mathrm{inf}~|\hat{\mu}(t)|=0~?$$

Edit : To be more precise, $\mu$ is continuous singular, so does not have a discret part (https://en.wikipedia.org/wiki/Lebesgue%27s_decomposition_theorem)

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