Geodesic
Two multidimensional analogues of the F.~and M.~Riesz theorem
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 164-176

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A new collection of Riesz sets for $\mathbb R^n$ and $\mathbb Z^n$ is exhibitize. The results are new even in $1$-dimensional case. An asymptotic estimate of Fourier multipliers on finite measures in $H^{1,\infty}$ space used for the proof is interesting in itself. 
@article{ZNSL_1998_255_a10,
     author = {M. M. Roginskaya},
     title = {Two multidimensional analogues of the {F.~and} {M.~Riesz} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {164--176},
     publisher = {mathdoc},
     volume = {255},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a10/}
}
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M. M. Roginskaya. Two multidimensional analogues of the F.~and M.~Riesz theorem. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 26, Tome 255 (1998), pp. 164-176. http://geodesic.mathdoc.fr/item/ZNSL_1998_255_a10/