Geodesic
On a uniqueness theorem for functions with a sparse spectrum
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 7-14

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We present an example of a set $\Lambda\in\mathbb Z$ satisfying the following two conditions: 1) there exists a nonzero positive singular measure on the unit circle $\mathbb T$ with spectrum in $\Lambda$; 2) if the spectrum of $f\in L^1(\mathbb T)$ is contained in $\Lambda$ and $f$ vanishes on a set of positive measure, then $f=0$. 
@article{ZNSL_1997_247_a0,
     author = {A. B. Aleksandrov},
     title = {On a uniqueness theorem for functions with a sparse spectrum},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {7--14},
     publisher = {mathdoc},
     volume = {247},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_247_a0/}
}
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A. B. Aleksandrov. On a uniqueness theorem for functions with a sparse spectrum. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 7-14. http://geodesic.mathdoc.fr/item/ZNSL_1997_247_a0/