A function f(x) is linear if f(x+y)=f(x)+f(y) for all pairs (x,y). Suppose f is "a bit linear" -- say, f(x+y)=f(x)+f(y) for 1% of pairs (x,y). What can you say about f? Must it be closely related to an actually linear function? If so, how closely?

This question turns out to be equivalent to asking for good quantitative bounds in the Freiman--Ruzsa theorem, a foundational result in additive combinatorics. Marton gave a formulation, equivalent to the statement above, which she conjectured should have polynomial bounds. I will outline a recent proof of this conjecture.

Joint work with Timothy Gowers, Ben Green and Terence Tao.