I am writing a short survey on connections between additive combinatorics and computer science for SIGACT News and I have been wondering about the “history” of the connections. (I will be writing as little as possible about history in the SIGACT article, because I don’t have the time to research it carefully, but if readers help me out, I would write about it in a longer article that I plan to write sometime next year.)
Now, of course, there is always that Russian paper from the 1950s . . .
Then there is the Coppersmith-Winograd matrix multiplication algorithm, which uses Behrend’s construction of large sets of integers with no length-3 arithmetic progressions. (They don’t need the full strength of Behrend’s construction, and the weaker construction of Salem and Spencer suffices.) As far as I know, this was an isolated application.
My understanding (which is quite possibly mistaken) is that there are three main threads to the current interactions: one concerning the Szemeredi Regularity Lemma and the “triangle removal lemma” of Ruzsa and Szemeredi; one concerning “sum-product theorems” and the Kakeya problem; and one concerning the Gowers uniformity norms. The earliest connection is the first one, and, depending how one is counting, started in 1992 or 2001, and is due, respectively, to Alon et al. or just to Alon.