# Alexander Chervov MO’s Question: Noteworthy-Achievements-In-And-Around-2010

Alexander Chervov asked over Mathoverflow about Noteworthy results in and around 2010  and some interesting results were offered in the answers. If you would like to mention additional results you can comment on them here. The only requirement is to explain what the result says and give links if possible.

1. Two that particularly struck me (and many other people) were the Guth-Katz solution of the Erdos distance problem and Tom Sanders’s results on Freiman’s theorem and Roth’s theorem. It hardly feels necessary to state these results, since they’ve been discussed a great deal, including here on this blog, but since those are the rules, here goes. Guth and Katz proved that any $n$ points in the plane must give rise to at least $n^{1-o(1)}$ distinct distances, and Sanders proved (amongst other things) that the largest density of a subset of $\{1,2,\dots,n\}$ that does not contain an AP of length 3 is $1/\log n$, to within a power of $\log\log n$.