Rapidly growing functions with empty spectrum and a gap in the support

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.