Let \(\{\varphi _{j}(x,y)\}_{j =1}^{\infty }\) be a uniformly bounded orthonormal system on \([0,1]^2\) , \(\{A_{n}\}_{n=1}^{\infty }\) be sequence of bounded subsets of \(\mathbb {Z}_{+}=\{1, 2, \dots \}\) and \(\{N_n\}\) denote the number of elements in \(A_n\) . Let also \(L_{A_n}(x,y)\) and \(S_{A_{n}}(f; x,y)\) denote, respectively, the Lebesgue function and the partial sum of the Fourier series of a function f with respect to the system \(\{\varphi _{j}(x,y)\}\) corresponding to the indices from \(A_n\) . It is proved that if the limit of the sequence \(\{L_{A_n}(x,y)/\log N_n\}\) is \(\infty \) almost everywhere on \([0,1]^2\) , then there exists an integrable on \([0,1]^2\) function h and a strictly increasing sequence \(\{n_k\}\) of positive integers, such that \( \lim _{k\rightarrow \infty } \mid S_{A_{n_k}}(h;x,y)\mid =\infty \) almost everywhere.
