The general question we are interested in is, given a space X, to find subsets S as small as possible with the property that every Lipschitz function on X has a point of differentiability in S (in which case we say S is a universal differentiability set). The Rademacher theorem says that Lipschitz functions on finite-dimensional spaces are differentiable almost everywhere. Hence every subset S of positive Lebesgue measure is a universal differentiabililty set. It turns out that not only there exist null universal differentiability sets but one can choose them to be compact null sets of upper box-counting dimension 1. In this talk, I will speak about the geometry of such sets, the general approach to constructing such sets and possible applications to open problems.
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