Multiplicative inequalities for the integral norm and problems of almost everywhere convergence of Fourier series

New results in the Littlewood-Paley theory are established and the method of obtaining sharp multiplicative inequalities for the integral norm of trigonometric and power series which permits to take into account both density and number-theoretic and combinatorial properties of spectrum is eleborated. The method is applied to series with spectra of power density including quadratic spectrum. The fundamental Kolmogorov's theorem is generalized to arbitrary bounded biorthogonal systems that are defined on a measurable space. Sharp lower estimates at the point and on the set of positive measure for arithmetical means from the symmetrized Lebesgue functions of biorthogonal systems are found. New results on the problem of convergence and unconditional convergence almost everywhere of Fourier series with respect to the Walsh system and multiplicative systems are obtained.