Asymptotics and zeros of the Fourier-Stieltjes transform of Cantor-Lebesgue finite measures (Q1360770)

We recall that a Cantor-Lebesgue measure \(\mu\) is a singular measure, whose support is a perfect set in some segment \([-a,a]\) with a constant dissection ratio \(\xi\), \(0<\xi<1/2\). The Fourier-Stieltjes transform of \(\mu\) is defined by \[ \mu(z):= \int^a_{-a} e^{izt}dF_\mu(t),\quad z\in\mathbb{C}, \] where \(F_\mu(t)\) is the distribution function of \(\mu\). The main result states, without proof, that the zeros \(z_k= x_k+ iy_k\) of \(\mu(z)\) are such that \[ |y_k|< C_\xi\ln^2|x_k| \] for sufficiently large \(k\). The example of the classical Cantor ``ladder'' (\(\xi:= 1/3\)) shows that this inequality is exact with respect to the exponent of the logarithm.