Extremal sets for Freiman's theorem

Abstract: The well-known theorem of Freiman states that sets of integers with small doubling are dense subsets of $d$--dimensional arithmetic progressions. In connection with this theorem, Freiman conjectured a precise upper bound on the volume of a finite $d$--dimensional set $A$ in terms of the cardinality of $A$ and of the sumset $A+A$. A set $A\subset {\mathbb Z}^d$ is $d$--dimensional if it is not contained in a hyperplane. Its volume is the smallest number of lattice points in the convex hull of a set $B$ that is Freiman isomorphic to $A$. The conjecture is equivalent to saying that the extremal sets for this problem are long simplices, consisting of a $d$--dimensional simplex and an extremal $1$--dimensional set in one of the dimensions. In this talk we will discuss a proof of the conjecture for a wide class of sets called chains. A finite set is a chain if there is an ordering of its elements such that initial segments in this ordering are extremal.

Joint work with G.A. Freiman.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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