Salvatore Tringali <salvo.tringali@gmail.com>





  Title : Minkowski duality, measured structures, and inequalities of Cauchy-Davenport type
  1. Minkowski duality is an abstraction of the common observation that, say, the structure of a group (A,+) implies a "derived" structure on the powerset of A (even though poorer than the original structure): The "derived" structure is a monoid (so unital and associative), but not a group, and not even cancellative (this is, for instance, the subject of Hamidoune's work on acyclic sgrps, which has served as a motivation for some ideas). Similar considerations apply to other "structures", such as rings, vector spaces, finitely generated groups, etc. And the basic question is: How to extend the "philosophy" on which these elementary cases are based to define a "derived structure" for any given structure? But for this to make sense we need to answer another preliminary question, namely: What is a structure, at all? And here is where model theory enters the scene. Of course, we cannot do it in a case-by-case fashion, since there are infinitely many types of structures, and I would say that it's not even true that only a finite amount of them is considered in "real applications".
  2. Minkowski duality is then used to define measured structures, which are in turn an extension of measured spaces in the sense of describing an "interplay" between a measure and an algebraic structure. Some examples are already there in the literature (I'm thinking, e.g., of Haar measures and locally compact Hausdorff topological groups), but to the best of my knowledge there is nothing like a general framework where to state (and possibly prove, once and for all!) something like an abstract "Cauchy-Davenport theorem", to the extent of generalizing (as many as possible) inequalities of Cauchy-Davenport type, coming from different particular settings (some of which I mentioned in the previous email), in the same way as the Cauchy-Davenport theorem for semigroups does with Chowla's, Pillai's, Kemperman's and others' theorems.

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Salvatore TRINGALI

Marie Curie fellow under project ApProCEM
Grant Agreement No. 276487
Université Pierre et Marie Curie
4 place Jussieu, 75005 Paris (FR)

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