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SUMMARY:Misha Rudnev (University of Bristol)
DTSTART:20201104T160000Z
DTEND:20201104T170000Z
DTSTAMP:20260423T024745Z
UID:hnts/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/hnts/12/">On
  convexity and sumsets</a>\nby Misha Rudnev (University of Bristol) as par
 t of Heilbronn number theory seminar\n\n\nAbstract\nA finite set $A$ of $n
 $ reals is convex if the sequence of neighbouring differences is strictly 
 monotone. Erdös suggested that that the set of squares of the first $n$ i
 ntegers may constitute the extremal case\, namely that for any convex $A$\
 , $|A+A| > n^{2-o(1)}$. The question is still open\, we'll review some par
 tial progress.\n\nWhat about $k- $fold sums $A+A+A+...$? In the case of sq
 uares\, they stop growing after $k=2$\, and for $k$th powers they grow up 
 to $n^k$. In a joint work with Brandon Hanson and Olly Roche-Newton we sho
 w\, using elementary methods\, that if $A=f([n])$\, where $f$ is a real fu
 nction with $k-1$ strictly monotone derivatives\, taking sufficiently many
  sums does lead to growth up to $n^{k-o(1)}$. We generalise this by replac
 ing the interval $[n]$ versus $f([n])$ by any set with small additive doub
 ling versus its image by $f$\, which enables us to apply this to sum-produ
 ct type questions.\n
LOCATION:https://researchseminars.org/talk/hnts/12/
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