The Kakeya problem asks whether a set in Rd 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.