Geodesic
A remark on indicator functions with gaps in the spectrum
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 108-115

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Developing a recent result of F. Nazarov and A. Olevskii, we show that for every subset $a$ of $\mathbb R$ of finite measure and every $\varepsilon>0$, there exists $b\subset\mathbb R$ with $|b|=|a|$ and $|(b\setminus a)\cup (a\setminus b)|\le\varepsilon$ such that the spectrum of $\chi_b$ is fairly thin. A generalization to locally compact Abelian groups is also provided. 
@article{ZNSL_2018_467_a9,
     author = {S. V. Kislyakov},
     title = {A remark on indicator functions with gaps in the spectrum},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {108--115},
     publisher = {mathdoc},
     volume = {467},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_467_a9/}
}
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S. V. Kislyakov. A remark on indicator functions with gaps in the spectrum. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 108-115. http://geodesic.mathdoc.fr/item/ZNSL_2018_467_a9/