Geodesic
Rapidly growing functions with empty spectrum and a gap in the support
Algebra i analiz, Tome 5 (1993) no. 3, pp. 77-99 Cet article a éte moissonné depuis la source Math-Net.Ru

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The theorem of the brothers Riesz says that certain bounded measures on $\mathbb R$ and Lebesgue measure have the same null sets. Over the years this theorem has been extended in a variety of ways. Recently, F. Forelli [F], showed that it holds for measures whose variation does not grow too fast. Here it is shown that the result of Forelli is sharp. More precisely, it is shown that for any sufficiently regular function $V\colon[0,+\infty)\to[1,+\infty)$ such $\int_0^\infty\frac{\log V(x)}{1+x^2}dx=\infty$ there exists a measure $\mu$, $\mathrm{Var}_{[-x,x]}|\mu|\le V(x)$, which has empty spectrum and which is not mutually absolutely continuous with Lebesgue measure.
Keywords: Theorem of the brothers Riesz
Mots-clés : Fourier–Carleman transform. 
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     author = {A. L. Volberg},
     title = {Rapidly growing functions with empty spectrum and a~gap in the support},
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A. L. Volberg. Rapidly growing functions with empty spectrum and a gap in the support. Algebra i analiz, Tome 5 (1993) no. 3, pp. 77-99. http://geodesic.mathdoc.fr/item/AA_1993_5_3_a1/