Tamas Keleti: How I could not solve the Besicovitch/Kakeya problem (yet)

By a Besicovitch set in R^n we mean a set that contains a unit line segment in every direction. The Kakeya conjecture states that such a set must have Minkowski and Hausdorff dimension n. We will see how different hoped results would imply some versions of this conjecture. All of these hoped results we can prove only for cases which are not relevant in the original problem (but interesting in itself).

I personally believe only in the Minkowski version of the conjecture. A possible way to disprove the Hausdorff version would be to show that a typical (in Baire category sense) Besicovitch set is a counter-example. It turns out that if a counter-example exists then it is typical. I will also mention some similar type of results in which the category argument gives examples of minimal Hausdorff dimension.