On the growth rate of arbitrary sequences of double rectangular Fourier sums

The theorem is proved that an arbitrary sequence $\{S_{m_k,n_k}(f,x,y)\} _{k=1}^\infty$ of double rectangular Fourier sums of any function from the class $L(\ln^+L)^2([0,2\pi)^2)$ satisfies almost everywhere the relation $S_{m_k,n_k}(f,x,y)=o(\ln k)$.