The Fourier transform of the characteristic function of a set, vanishing on an interval

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.
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