We give improved lower bounds on the size of Kakeya and Nikodym sets over Fq. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in Fq, and show that this conjecture implies an optimal bound on the size of a Nikodym set. Finally, we study the notion of a weak Nikodym set and give improved, and in some special cases optimal, bounds fo...

Semilinear multivalued equations are considered, in separable Banach spaces with the Radon-Nikodym property. An effective criterion for the existence of solutions to the associated Floquet boundary value problem is showed. Its proof is obtained combining a continuation principle with a Liapunov-like technique and a Scorza-Dragoni type theorem. A strictly localized transversality condition is as...

We define the (w∗-) boundedness property and the (w∗-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called (w∗-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of tw...

We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map u : X → R such that for every interval I ⊂ R, the connected components of u−1(I) have diameter ≤ const ·diam(I), then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples [Laa00] bilipschitz embed in L1, though they do not embed i...