Marstrand's Approximate Independence of Sets and Strong Differentiation of the Integral

Authors

  • Raquel Cabral

DOI:

https://doi.org/10.7146/math.scand.a-24186

Abstract

A constructive proof is given for the existence of a function belonging to the product Hardy space H1(R×R) and the Orlicz space L(logL)ϵ(R2) 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.

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Published

2016-08-19

How to Cite

Cabral, R. (2016). Marstrand’s Approximate Independence of Sets and Strong Differentiation of the Integral. MATHEMATICA SCANDINAVICA, 119(1), 92–112. https://doi.org/10.7146/math.scand.a-24186

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Articles