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SUMMARY:Chi Hoi Yip (University of British Columbia)
DTSTART:20220524T200000Z
DTEND:20220524T202500Z
DTSTAMP:20260423T011438Z
UID:CANT2022/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/12/
 ">Asymptotics for the number of directions determined by $[n] \\times [n]$
  in $\\mathbb{F}_p^2$</a>\nby Chi Hoi Yip (University of British Columbia)
  as part of Combinatorial and additive number theory (CANT 2022)\n\n\nAbst
 ract\nLet $p$ be a prime and $n$ a positive integer such that $\\sqrt{\\fr
 ac p2} + 1 \\leq n \\leq \\sqrt{p}$. For any arithmetic progression $A$ of
  length $n$ in $\\mathbb{F}_p$\, we establish an asymptotic formula for th
 e number of directions determined by $A \\times A \\subset \\mathbb{F}_p^2
 $. The key idea is to reduce the problem to counting the number of solutio
 ns to the bilinear Diophantine equation $ad+bc=p$ in variables $1\\le a\,b
 \,c\,d\\le n$\; our asymptotic formula for the number of solutions is of i
 ndependent interest. \n\nJoint work with Greg Martin and Ethan White.\n
LOCATION:https://researchseminars.org/talk/CANT2022/12/
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