On the complete characterization of differentiation sets of integrals

Abstract

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