On the complete characterization of differentiation sets of integrals

Abstract

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}$).