On the convergence of multiple Haar series

We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: $L(\ln^+L)^{n-1}$ is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.