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SUMMARY:Sophie Huczynska (University of St. Andrews\,)
DTSTART:20250520T150000Z
DTEND:20250520T152500Z
DTSTAMP:20260423T010457Z
UID:CANT2025/15
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/15/
 ">Additive triples in groups of odd prime order</a>\nby Sophie Huczynska (
 University of St. Andrews\,) as part of Combinatorial and additive number 
 theory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Science Cente
 r (4th floor).\n\nAbstract\nFor a subset $A$ of an additive group $G$\,  a
  Schur triple in $A$ is a triple of the form $(a\,b\,a+b) \\in A^3$.  Deno
 te by $r(A)$ the number of Schur triples of $A$\; the behaviour of $r(A)$ 
 as $A$ ranges over subsets of a group $G$ has been studied by various auth
 ors. When $r(A)=0$\, $A$ is sum-free. The question of minimum and maximum 
 $r(A)$ for $A$ of fixed size in $\\mathbb{Z}_p$ was resolved by Huczynska\
 , Mullen and Yucas (2009) and independently by Samotij and Sudakov (2016).
   Several generalisations of the Schur triple problem have received attent
 ion.  In this talk\, I will present recent work (with Jonathan Jedwab and 
 Laura Johnson) on the generalisation to triples $(a\,b\,a+b) \\in A \\time
 s B \\times B$\, where $A\,B \\subseteq \\mathbb{Z}_p$.  Denote by $r(A\,B
 \,B)$ the number of triples of this form\; we obtain a precise description
  of its full spectrum of values and show constructively that each value in
  this spectrum can be realised when $B$ is an interval of consecutive elem
 ents in $\\mathbb{Z}_p$.\n
LOCATION:https://researchseminars.org/talk/CANT2025/15/
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