A note -- if you're linking to arXiv, it's better to link to the abstract (http://arxiv.org/abs/1509.05363) rather than directly to the PDF. From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to do things like see different versions of the paper, search for other things by the same authors, etc. Thank you!
The original Nature article had some very good background, including a link to the blog comment that gave Tao the impetus to explore a connection that led to this discovery (and which Tao credited in the paper):