Geodesic
The Fourier transform of the characteristic function of a set, vanishing on an interval
Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 397-410 Cet article a éte moissonné depuis la source Math-Net.Ru

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A set $E$ with $0<\operatorname{mes}E<+\infty$ is constructed for which the Fourier transform of its characteristic function vanishes on an interval. The set is the union of a sequence of intervals whose lengths can be estimated asymptotically above and below. The construction is based on an infinite-dimensional version of the implicit function theorem. Bibiography: 6 titles. 
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     title = {The {Fourier} transform of the characteristic function of a~set, vanishing on an~interval},
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P. P. Kargaev. The Fourier transform of the characteristic function of a set, vanishing on an interval. Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 397-410. http://geodesic.mathdoc.fr/item/SM_1983_45_3_a4/

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