The Kakeya problem asks whether a set in R^{d}
containing a line in every direction must have Hausdorff
dimension d. In three dimensions the best known lower bound
for the dimension was, for a while, 2.5, by an argument of
Wolff, building on work of Bourgain. Then Katz, Laba and
Tao improved this to 2.5 + 10^{-10}. Why was this
an interesting improvement?

The answer is that there is an example related to the
Heisenberg group of a set which has 2.5 dimensions and has
properties quite similar to having a line in every direction.
Indeed, these properties are similar enough that Wolff's
methods could not possibly have yielded a better bound,
since they would have applied equally well to this example
where the bound really is 2.5. Thus, Katz, Laba and Tao
had to find some way of distinguishing the genuine Kakeya
problem from this `fake' Kakeya problem. That was very hard,
so the 10^{-10} was a genuine improvement.