Jayadev Athreya was here last week and reminded me about this notion of “winning sets,” which I learned about from Howie Masur — originally, one of the many contributions of Wolfgang Schmidt.
Here’s a paper by Curt McMullen introducing a somewhat stronger notion, “absolute winning.”
Anyway: a winning set (or an absolute winning set) in R^n is “big” in some sense. In particular, it has to have full Hausdorff dimension, but it doesn’t have to have positive measure.
Kakeya sets (subsets of R^n containing a unit line segment in every direction) can have measure zero, by the Besicovitch construction, and are conjectured (when n=2, known) to have Hausdorff dimension n. So should we expect these sets to be winning? Are Besicovitch sets winning?
I have no reason to need to know. I just think these refined classifications of sets which are measure 0 yet still “large” are very interesting. And for all I know, maybe there are sets where the easiest way to prove they have full Hausdorff dimension is to prove they’re winning!