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SUMMARY:Junxuan Shen (California Institute of Technology)
DTSTART:20220526T183000Z
DTEND:20220526T185500Z
DTSTAMP:20260423T011340Z
UID:CANT2022/40
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/40/
 ">The structural incidence problem for cartesian products</a>\nby Junxuan 
 Shen (California Institute of Technology) as part of Combinatorial and add
 itive number theory (CANT 2022)\n\n\nAbstract\nWe prove new structural res
 ults for point-line incidences. An incidence is a pair of one point and on
 e line\, where the point is on the line. The Szemer\\'{e}di-Trotter theore
 m states that $n$ points and $n$ lines form $O(n^{4/3})$ incidences. This 
 bound has been used to obtain many results in combinatorics\, number theor
 y\, harmonic analysis\, and more. While the Szemer\\'{e}di-Trotter bound h
 as been known for several decades\, the structural problem remains wide-op
 en. This problem asks to characterize the point-line configurations with $
 \\Theta(n^{4/3})$ incidences. \nWe prove that when the point set $\\mathca
 l{P}$ is a Cartesian product where only one axis of it behaves like a latt
 ice\, the line set must contain many families of parallel lines to achieve
  the maximal incidence bound.\n\nTheorem: \nConsider $1/3<\\alpha<2/3$. Le
 t $A\,B\\subset\\RR$ satisfy that $A=\\{1\,2\,\\cdots\, n^{\\alpha}\\}$ an
 d $|B|=n^{1-\\alpha}$. Let $\\mathcal{L}$ be a set of $n$ lines in $\\RR^2
 $\, such that $I(A\\times B\,\\mathcal{L})=\\Theta(n^{4/3})$. Then $\\math
 cal{L}$ contains $\\Omega(n^{1-\\beta}/\\log n)$ disjoint families of $\\T
 heta(n^{\\beta})$ parallel lines for $1-2\\alpha\\le\\beta\\le 2/3$.\n\nWh
 en $\\alpha<1/3$ or $\\alpha>2/3$\, it is impossible to have $\\Theta(n^{4
 /3})$ incidences. We also completely characterize the line set when the po
 int set is a lattice.\n\nJoint work with Adam Sheffer. \\\\\n
LOCATION:https://researchseminars.org/talk/CANT2022/40/
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