On the complete characterization of differentiation sets of integrals

Let $B_\theta$ be the family of rectangles in the plane $R^2$, having slope $\theta$ with the abscissa. We say a set of slopes $\Theta$ is $D$-set if there exists a function $f\in L(R^2)$, such that the basis $B_\theta$ differentiates integral of $f$ if $\theta\not\in\Theta$ and $\bar D_\theta f(x)=\infty$ almost everywhere if $\theta\in\Theta$. If the condition $\bar D_\theta f(x)=\infty$ holds on a set of positive measure (instead of a.e.) we shall say it is $WD$-set. It is proved, that $\Theta$ is $D$-set($WD$-set) if and only if it is $G_\delta$($G_{\delta\sigma}$).