SPEAKER: Terence Tao
TITLE: Long arithmetic progressions in the primes
ABSTRACT:
We present some joint work with Ben Green on counting the number of arithmetic
progressions of length k in the integer primes, and in particular showing
that for each k there are infinitely many of these progressions. One key
idea is to view the primes not as a sparse subset of Z, but rather as a dense
subset of "almost primes", which (after some local obstructions from small
divisors are eliminated) are distributed "pseudorandomly" in the integers,
thanks to some recent work of Goldston and Yildirim. One can then transfer
Szemeredi's theorem (dense subsets of integers contain arbitrarily long progressions)
to this setting and obtain arbitrarily long progressions of primes. More
recently, we have been able to transfer the more Fourier analytic analysis
of Gowers concerning the k=4 case of Szemeredi's theorem, and thus obtain
asymptotics for the number of progressions of primes for length 4. (The
case k=3 was already known to van der Corput).