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SUMMARY:Eshita Mazumdar (ISI Bangalore)
DTSTART:20200930T053000Z
DTEND:20200930T063000Z
DTSTAMP:20260423T021351Z
UID:CATGT/10
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CATGT/10/">I
 terated sumsets and Hilbert functions</a>\nby Eshita Mazumdar (ISI Bangalo
 re) as part of Applications of Combinatorics in Algebra\, Topology and Gra
 ph Theory\n\n\nAbstract\nLet $A$ be a finite subset of an abelian group $(
 G\,+)$. Let $h \\ge 2$ be an integer. If $|A| \\ge 2$ and the cardinality 
 $|hA|$ of the $h$-fold iterated sumset $hA=A+\\dots+A$ is known\, what can
  one say about $|(h-1)A|$ and $|(h+1)A|$? It is known that $$|(h-1)A| \\ge
  |hA|^{(h-1)/h}\,$$ a consequence of Pl\\"unnecke's inequality. we improve
 d this bound with a new approach. Namely\, we model the sequence $|hA|_{h 
 \\ge 0}$ with the Hilbert function of a standard graded algebra. We then a
 pply Macaulay's 1927 theorem on the growth of Hilbert functions\, and more
  specifically a recent condensed version of it. Our bound implies $$|(h-1)
 A| \\ge \\theta(x\,h)\\hspace{0.4mm}|hA|^{(h-1)/h}$$ for some factor $\\th
 eta(x\,h) > 1$\, where $x$ is a real number closely linked to $|hA|$. More
 over\, we show that $\\theta(x\,h)$ asymptotically tends to $e\\approx 2.7
 18$ as $|A|$ grows and $h$ lies in a suitable range varying with $|A|$. Th
 is is a joint work with Prof. Shalom Eliahou.\n
LOCATION:https://researchseminars.org/talk/CATGT/10/
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