Щхе existence Fragmen–Lindelof function and some conditions of quasi-analyticity

Let $E\subset\mathbb R^n$, $E=\bar E$, $\Omega=\mathbb R^{n+1}\setminus E$. Positive harmonic functions in $\Omega$ vanishing on $E$, form the cone $\mathcal P_E$. It is known, that $1\leqslant\dim\mathcal P_E\leqslant2$. It is proved, that $\int_{\mathbb R^n}\frac{\rho(x, E)}{(1+x^2)^{\frac{n+1}2}}=+\infty\Rightarrow\dim \mathcal P_E=1$ ($\rho(x, E)=\inf_{t\in E}|x-t|$). The connection between $\dim\mathcal P_E$ and the existence of a non-zero measure on $E$ whose Fourier transform vanishes pn an interval is investigated. In the case $n=1$ it is proved, that $\int_{C_E}\frac{dt}{1+|t|}<+\infty\Rightarrow\dim \mathcal P_E=2$.