On complete characterization of coefficients of a.e. converging orthogonal series

We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1} a_n\Phi_n$ converges a.e. for any orthonormal system $(\Phi_n)$ in any $L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |a_m|^2; n\geq 1\}$ and its information function $$h_B(t) = -\log_3(\beta-\alpha)$$ for $t\in (\alpha, \beta]$, $[\alpha, \beta]\cap B =\{\alpha, \beta\}.$