Let G1, . . ., Gk be vector spaces over a prime field 𝔽p. We say that a map Φ defined on a subset of the product G1 × . . . × Gk is a Freiman multi-homomorphism if Φ is a Freiman homomorphism in every principal direction (i.e. when xi in Gi is fixed for each i except one direction d, the map that sends element xd to Φ(x1,…, xk) is a Freiman homomorphism, where we allow those xd for which (x1,…, xk) is in the domain of Φ). It turns out that a Freiman multi-homomorphism defined on a dense subset of G1 × . . . × Gk necessarily coincides with a global multiaffine map at many points. In this talk we discuss the proof of this fact and some applications, which include a quantitative inverse theorem for the Gowers uniformity norms (in finite vector spaces in the case of large characteristic).
This is joint work with Tim Gowers.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
