On Fourier Coefficients of Lacunary Systems

We prove that the Zygmund space $L(\ln L)^{1/2}$ is the greatest one in the set of symmetric spaces $X$ for which any uniformly bounded orthonormal system of functions contains a sequence such that the corresponding space of Fourier coefficients $F(X)$ coincides with $\ell_2$. Similar results also hold for symmetric spaces located between the spaces $L(\ln L)^{1/2}$ and $L_1$.