We construct examples of uncountable compact subsets of complex numbers with the property that any Borel measure on the circle group with Fourier coefficients taking values in this set has a natural spectrum. For measures with Fourier coefficients tending to 0 we construct an open set with this property. We also give an example of a singular measure whose spectrum is contained in our set.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-2-2,
author = {Przemys\l aw Ohrysko and Micha\l\ Wojciechowski},
title = {On the relationships between Fourier-Stieltjes coefficients and spectra of measures},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {117-140},
zbl = {1295.43004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-2-2}
}
Przemysław Ohrysko; Michał Wojciechowski. On the relationships between Fourier-Stieltjes coefficients and spectra of measures. Studia Mathematica, Tome 223 (2014) pp. 117-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-2-2/