Marstrand’s Approximate Independence of Sets and Strong Differentiation of the Integral - Cabral, Raquel (MATHEMATICA SCANDINAVICA, 2016)
A constructive proof is given for the existence of a function belonging to the product Hardy space H^1(R × R) and the Orlicz space L(L)^ε(R^2) for all 0<ε <1, for all whose integral is not strongly differentiable almost everywhere on a set of positive measure. It consists of a modification of a non-negative function created by J. M. Marstrand. In addition, we generalize the claim concerning "approximately independent sets" that appears in his work in relation to hyperbolic-crosses. Our generalization, which holds for any sets with boundary of sufficiently low complexity in any Euclidean space, has a version of the second Borel-Cantelli Lemma as a corollary.