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

We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1} a_n\Phi_n$convergesa\mathrm{.}e\mathrm{.}foranyorthonormalsystem$(\Phi_n)$inany$L_2$-space\mathrm{.}Inourcriterion,weusetheset$B ={\sum_{m\geq n} |a_m|^2; n\geq 1}$anditsinformationfunction$$h_B(t)=-{\log }_3(\beta -\alpha )$$for$t\in (\alpha, \beta]$,$[\alpha, \beta]\cap B ={\alpha, \beta}.$