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SUMMARY:Oriol Serra (Universitat Politecnica de Catalunya\, Barcelona)
DTSTART:20200603T140000Z
DTEND:20200603T142500Z
DTSTAMP:20260423T011239Z
UID:CANT2020/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2020/31/
 ">Extremal sets for Freiman's theorem</a>\nby Oriol Serra (Universitat Pol
 itecnica de Catalunya\, Barcelona) as part of Combinatorial and additive n
 umber theory (CANT 2021)\n\n\nAbstract\nThe well-known theorem of Freiman 
 states that sets of integers with small doubling \nare dense subsets of $d
 $--dimensional arithmetic progressions.  \nIn connection with this theorem
 \,  Freiman conjectured a precise upper bound on the volume \nof a finite 
 $d$--dimensional set $A$ in terms of the cardinality of $A$ and  of the su
 mset $A+A$. \nA set $A\\subset {\\mathbb Z}^d$ is $d$--dimensional if it i
 s not contained in a hyperplane. \nIts volume is the smallest number of la
 ttice points in the convex hull of a set $B$ that is Freiman \nisomorphic 
 to $A$. The conjecture is equivalent to saying that the extremal sets for 
 this problem \nare long simplices\, consisting of a $d$--dimensional simpl
 ex and an extremal $1$--dimensional \nset in one of the dimensions. In thi
 s talk we will discuss a proof of the conjecture  for a wide class \nof se
 ts called chains.  A finite set is a chain if there is an ordering of its 
 elements such that initial \nsegments in this ordering are extremal. \n\nJ
 oint work with G.A. Freiman.\n
LOCATION:https://researchseminars.org/talk/CANT2020/31/
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