A complete characterization of R-sets in the theory of differentiation of integrals
Let R s be the family of open rectangles in the plane R 2 having slope s with the abscissa. We say a set of slopes S is R-set if there exists a function f ∈ L(R 2), such that the basis R s differentiates integral of f if s ∈ S, and D s f (x) = lim sup diam (R)→0,x∈R∈Rs 1 |R| R f = ∞ almost everywhere if s ∈ S. If the condition D s f (x) = ∞ holds on a set of positive measure (instead of a.e.) we shall say it is W R-set. It is proved, that S is a R-set(W R-set) if and only if it is G δ (G δσ).