Suppose that $f\in L^2(\mathbb{R})$ is non-zero and compactly supported. Then its Fourier transform $\widehat{f}\neq 0$ is analytic, and in particular the nodal set $\{\xi\in\mathbb{R}\,s.t.\,\widehat{f}(\xi)=0\}$ has Lebesgue measure zero. I'm wondering whether the same conclusion can be obtained by slightly relaxing the compactly supported condition. So my (a bit vague) question is the following:

**Are there some sufficient conditions (weaker than compactness) on the support of $f$ which guarantee that $\{\xi\in\mathbb{R}\,s.t.\,\widehat{f}(\xi)=0\}$ has Lebesgue measure zero?**

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