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

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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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     author = {P. P. Kargaev},
     title = {The {Fourier} transform of the characteristic function of a~set, vanishing on an~interval},
     journal = {Sbornik. Mathematics},
     pages = {397--410},
     publisher = {mathdoc},
     volume = {45},
     number = {3},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_45_3_a4/}
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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/