Lecture 13 (Apr 11)
Today we proved the Freiman-Ruzsa theorem. To do this, we developed the theory of Freiman isomorphisms, and proved the Ruzsa Modelling Lemma. More detailed notes can be found by clicking the button below.
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Lecture 12 (Apr 4)
Today we filled in a missing step in the proof of the Bohr-to-gAP proposition. To do so, we introduced lattices and discussed the Geometry of Numbers, in particular proving Minkowski's 1st theorem and motivating his 2nd theorem. Finally, we began discussing the notion of Freiman homomorphisms, which play a crucial role in the remainder of the proof of the Freiman-Ruzsa theorem. More detailed notes can be found by clicking the button below.
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Lecture 11 (Mar 28)
Today we proved the Bohr-to-gAP proposition: any low-dimensional Bohr set contains a large, low-dimensional proper gAP. More detailed notes can be found by clicking the button below.
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Lecture 10 (Mar 21)
Today we used Fourier analysis to prove Bogolyubov's Lemma that for every large subset A of Z/NZ, 2A-2A contains a Bohr set of small dimension. We then discussed the size of this Bohr set. For more detailed notes (by Gal), click the button below.
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Lecture 9 (Mar 14)
Today we reviewed Fourier analysis, in particular the theory over a finite abelian group. We then gave some examples of how it could be used to study problems in number theory. More detailed notes will be posted soon.
Lecture 9 notes will be posted
Lecture 8 (Mar 7)
Today we started in on the proof of Freiman-Ruzsa. As a first step, we reduced the problem to an easier one: rather than looking for a small gAP containing A, we look for a large gAP contained in mA-nA for some non-negative integers m and n. This is an easier problem, because the structure of mA-nA becomes smoother and smoother as m and n get larger. For more detailed notes (by Gal), click the button below.
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Lecture 7 (Feb 28)
We began by formally concluding Petridis' proof of Pluennecke-Ruzsa, formalizing a lemma we'd proved collaboratively in class last lecture. Next, we proved a lovely theorem of Ruzsa characterizing sets of small doubling in groups of bounded torsion. Finally, we discussed the Polynomial Freiman-Ruzsa Conjecture. For more detailed notes (by Gal), click the button below.
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Lecture 6 (Feb 14)
We began by discussing some connections between sumsets and difference sets, in particular stating several (potentially open) questions. This led us to compare the two notions of doubling, and how they relate to one another. This in turn inspired us to define and develop the Ruzsa distance. We then gave Petridis' proof of the Plünnecke-Ruzsa up to a key lemma, which we collaboratively proved. For more detailed notes (by Gal), click the button below.
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Lecture 5 (Feb 7)
Guest lecture by Lluis Vena, continued. Summary to be posted soon.
Lecture 4 (Jan 31)
Guest lecture by Lluis Vena. Summary to be posted soon.
Lecture 3 (Jan 24)
We concluded the proof of Solymosi's theorem, switched viewpoints to general abelian groups, enunciated an interpretation of what additive combinatorics is about, and sketched a proof of a result due to Laba. For more detailed notes (by Gal), click the button below.
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Lecture 2 (Jan 17)
We discussed the Multiplication Table problem from last time, as well as starting the proof of Solymosi's theorem. For more detailed notes (by Gal), click the button below.
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Lecture 1 (Jan 10)
We discussed administrative matters, notions of set addition and multiplication, the Erdos-Szemeredi conjecture, Solymosi's theorem, the Freiman-Ruzsa theorem, and related matters. For more detailed notes, click the button below.
