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DTSTART:20240310T070000
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DTSTART:20231105T060000
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DTSTART;TZID=America/Toronto:20240923T113000
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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/algebraic-gr
 aph-theory-seoyoung-kim
SUMMARY:Algebraic Graph Theory - Seoyoung Kim
CLASS:PUBLIC
DESCRIPTION:TITLE: Diophantine tuples\, graphs\, and additive combinatorics
 \n\nSPEAKER:\n Seoyoung Kim\n\nAFFILIATION:\n University of Göttingen\n\n
 LOCATION:\n Please contact Sabrina Lato for Zoom link.\n\nABSTRACT: In 
 this talk\, we investigate the multiplicative structure\nof a shifted mult
 iplicative subgroup and its connections with additive\ncombinatorics and t
 he theory of Diophantine tuples and Diophantine\ngraphs. First\, we show 
 that if a nontrivial shift of a multiplicative\nsubgroup $G$ contains a pr
 oduct set $AB$\, then $|A||B|$ is essentially\nbounded by $|G|$\, refining
  a well-known consequence of a classical\nresult by Vinogradov. Second\, w
 e provide a sharper upper bound of\n$M_k(n)$\, or the clique number of Di
 ophantine graphs\, which is the\nlargest size of a set such that each pair
 wise product of its elements\nis $n$ less than a $k$-th power\, refining t
 he recent result of Dixit\,\nKim\, and Murty.  One main ingredient in our
  proof is the first\nnon-trivial upper bound on the maximum size of a gene
 ralized\nDiophantine tuple over a finite field. In addition\, we determine
  the\nmaximum size of an infinite family of generalized Diophantine tuples
 \nover finite fields with square order\, which is of independent\ninterest
 .  \n\nWe also present a significant progress towards a conjecture of\nS\
 \'{a}rk\\\"{o}zy on the multiplicative decompositions of shifted\nmultipli
 cative subgroups. In particular\, we prove that for almost all\nprimes $p$
 \, the set $\\{x^2-1: x \\in \\F_p^*\\} \\setminus \\{0\\}$ cannot\nbe dec
 omposed as the product of two sets in $\\F_p$ non-trivially.
DTSTAMP:20260403T083056Z
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