Introduction to additive combinatorics lecture 5.8 --- Freiman homomorphisms and isomorphisms.

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The notion of a Freiman homomorphism and the closely related notion of a Freiman isomorphism are fundamental concepts in additive combinatorics. Here I explain what they are and prove a lemma that states that a subset A of F_p^N such that kA - kA is not too large is "k-isomorphic" to a subset of F_p^n for an n such that p^n is not much larger than A. (The precise bound obtained for n is that if |kA-kA| is at most C|A|, then p^{n-1} is less than C|A|.)

0:00 Introduction
1:26 Motivating example
4:59 Definition of a Freiman homomorphism of order k
9:48 Some basic facts about Freiman homomorphisms
16:27 Definition of a Freiman isomorphism of order k
18:23 Properties preserved by Freiman isomorphisms
24:57 Statement of the Ruzsa embedding lemma for F_p^N
28:07 Proof of the lemma
37:40 Very brief preview of the next video

Timothy Gowers

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Hi Prof. Gowers, in the video mention is made of an examples sheet. Would it be possible for these to be made publicly available, like the notes are? Apologies if I've missed the link somewhere.

marcelgoh

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