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SUMMARY:Sergei Konyagin (Russia)
DTSTART:20260714T133000Z
DTEND:20260714T142500Z
DTSTAMP:20260805T044339Z
UID:CANT2026/22
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/22/
 ">On Sidon sets with squares\, cubes and quartics in short intervals</a>\n
 by Sergei Konyagin (Russia) as part of Combinatorial and additive number t
 heory seminar (CANT 2026)\n\nLecture held in Science Center in the CUNY Gr
 aduate Center (4th floor).\n\nAbstract\nFor any positive integer $N$\, the
  equation  $\n x^3+y^3=z^3+t^3\,  \\quad\nx\,y\,z\,t\\in \\mathbb{N}\, \\q
 uad  \\{x\,y\\}\\not=\\{z\,t\\} $  has no solution  satisfying  $\n N\\le 
 x\,y\,z\,t <\nN+\\Bigl(\\frac{38}{3}N+\\frac{1297}{36}\\Bigr)^{1/2}+\\frac
 {19}{6}. $ \nThe strict\ninequality ``$<$" can not be substituted by ``$\\
 le$"\, that is\, there exist  infinitely many positive integers $N$ such t
 hat the equation has a solution\nwith   $\n N\\le x\,y\,z\,t \\le\nN+\\Big
 l(\\frac{38}{3}N+\\frac{1297}{36}\\Bigr)^{1/2}+\\frac{19}{6}. $   \n There
  is an absolute constant $c>0$ such that for any positive integer $N$\nthe
  equation  has a solution satisfying  $ N\\le x\,y\,z\,t \\le N+cN^{2/3}. 
 $  \n  For any $\\varepsilon>0$ there exist infinitely many positive integ
 ers $N$\nsuch that the equation  has no solution  satisfying   $ N\\le x\,
 y\,z\,t \\le N+N^{4/7-\\varepsilon}. $  \n  There is an absolute constant 
 $c>0$ such that for any positive integer $N$\nthe equation  \n$ x^4+y^4=z^
 4+t^4\,\\quad x\,y\,z\,t\\in\\mathbb{N}\, \\quad\n\\{x\,y\\}\\not=\\{z\,t\
 \}\, $ \\\\& \nhas no solution satisfying   \n$ N\\le x\,y\,z\,t \\le N+cN
 ^{3/5}. $  \n  There is an absolute constant $c>0$ such that for any posit
 ive integer $N$\nthis equation  has a solution satisfying   \n$ N\\le x\,y
 \,z\,t \\le N+cN^{12/13}. $   \nThe talk is based on a joint paper of the 
 speaker with\nM.~Z.~Garaev and F.~M.~Garayev.\n
LOCATION:https://researchseminars.org/talk/CANT2026/22/
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