In this talk I will try to describe (part of) an ongoing reasearch with D. Preiss and M. Csoernyei.
The starting point is a decomposition theorem for null sets (sets of measure zero) in the plane, which can be obtained as a direct consequence of a geometric version of the Erdos-Szekeres theorem.
I will outline some applications to problems in Geometric Measure Theory (and not only), and if time permits, I will finally address the issue of
extending these results to higher dimensions.