The majority of Green's research is in the fields of
analytic number theory and
additive combinatorics, but he also has results in
harmonic analysis and in
group theory. His best known theorem, proved jointly with his frequent collaborator
Terence Tao, states that there exist arbitrarily long
arithmetic progressions in the
prime numbers: this is now known as the
Green–Tao theorem. Amongst Green's early results in additive combinatorics are an improvement of a result of
Jean Bourgain of the size of arithmetic progressions in
sumsets, as well as a proof of the
Cameron–Erdős conjecture on sum-free sets of
natural numbers. He also proved an arithmetic regularity lemma for functions defined on the first N natural numbers, somewhat analogous to the
Szemerédi regularity lemma for graphs. From 2004–2010, in joint work with
Terence Tao and
Tamar Ziegler, he developed so-called
higher order Fourier analysis. This theory relates
Gowers norms with objects known as
nilsequences. The theory derives its name from these nilsequences, which play an analogous role to the role that
characters play in classical
Fourier analysis. Green and Tao used higher order Fourier analysis to present a new method for counting the number of solutions to simultaneous equations in certain sets of integers, including in the primes. This generalises the classical approach using
Hardy–Littlewood circle method. Many aspects of this theory, including the quantitative aspects of the inverse theorem for the Gowers norms, are still the subject of ongoing research. Green has also collaborated with
Emmanuel Breuillard on topics in group theory. In particular, jointly with
Terence Tao, they proved a structure theorem for
approximate groups, generalising the
Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has worked, jointly with
Kevin Ford and
Sean Eberhard, on the theory of the
symmetric group, in particular on what proportion of its elements fix a set of size k. Green and Tao also have a paper on algebraic
combinatorial geometry, resolving the Dirac-Motzkin conjecture (see
Sylvester–Gallai theorem). In particular they prove that, given any collection of n points in the plane that are not all
collinear, if n is large enough then there must exist at least n/2 lines in the plane containing exactly two of the points.
Kevin Ford, Ben Green,
Sergei Konyagin,
James Maynard and
Terence Tao, initially in two separate research groups and then in combination, improved the lower bound for the size of the longest gap between two consecutive primes of size at most X. The form of the previously best-known bound, essentially due to
Rankin, had not been improved for 76 years. More recently Green has considered questions in arithmetic
Ramsey theory. Together with
Tom Sanders he proved that, if a sufficiently large
finite field of prime order is coloured with a fixed number of colours, then the field has elements x,y such that x, y, x{+}y, xy all have the same colour. Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the
polynomial method to bound the size of subsets of a finite
vector space without solutions to
linear equations. He adapted these methods to prove, in function fields, a strong version of
Sárközy's theorem. == Awards and honours ==