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SUMMARY:A model-theoretic approach to Roth's theorem - Amador Martin-Pizar
 ro (Albert-Ludwigs-Universität Freiburg)
DTSTART:20250212T133000Z
DTEND:20250212T150000Z
UID:TALK225121@talks.cam.ac.uk
CONTACT:Julia Wolf
DESCRIPTION:The ultraproduct construction is a useful tool in model theory
  to study the asymptotic behavior of a class of structures. In the particu
 lar case of a class of finite groups\, the ultralimit of the normalized co
 unting measure yields a translation-invariant Keisler measure on internal 
 sets\, which has played a crucial role in the recent years in several appl
 ications of model-theoretic techniques to additive combinatorics. \n\nIn t
 his talk\, we present a model-theoretic result that resonates with Croot-S
 isask's almost periodicity technique for a general group equipped with a K
 eisler measure under some mild assumptions. We then show how to use this r
 esult to obtain\, via an ultrafilter construction\, a non-quantitative pro
 of of Roth’s theorem on arithmetic progressions of length three. The cor
 e idea of our model-theoretic version of almost periodicity is the stabili
 ty-like behaviour of a convolution of sets. We will not assume prior knowl
 edge of model theory for this talk.\n\nIn the first part of the talk\, aim
 ed at a general (non-logic) audience\, we will recall the ultraproduct con
 struction of finite groups\, as well as Łoś's theorem\, dense internal s
 ubsets and the main features of stable relations\, in order to briefly out
 line how to prove a non-quantitative version of Roth's theorem.\n\nThe sec
 ond part of the talk will focus on a more detailed explanation of some asp
 ects of the proofs\, in particular the notions of dense and random element
 s and their features. If time permits\, we will explain how some of these 
 techniques can be adapted to study the collection of starting points of ar
 ithmetic progressions in the primes and in the square-free integers.
LOCATION:MR4\, CMS
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