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SUMMARY:Thai Hoang Le (University of Mississippi)
DTSTART:20220526T173000Z
DTEND:20220526T175500Z
DTSTAMP:20260423T011340Z
UID:CANT2022/38
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/38/
 ">Bohr sets in sumsets in countable abelian groups</a>\nby Thai Hoang Le (
 University of Mississippi) as part of Combinatorial and additive number th
 eory (CANT 2022)\n\n\nAbstract\nA \\textit{Bohr set} in an abelian topolog
 ical group $G$ is a subset of the form\n\\[\nB(K\, \\epsilon) = \\{ g \\in
  G: |\\chi(g) - 1| < \\epsilon \\\, \\forall \\chi \\in K \\}\n\\]\nwhere 
 $K$ is a finite subset of the dual group $\\widehat{G}$. A classical theor
 em of Bogolyubov says that if $A \\subset \\mathbf{Z}$ has positive upper 
 density $\\delta$\, then $A+A-A-A$ contains a Bohr set $B(K\, \\epsilon)$ 
 where $|K|$ and $\\epsilon$ depend only on $\\delta$. While the same state
 ment for $A-A$ is not true (a result of K\\v{r}\\'i\\v{z})\, Bergelson and
  Ruzsa proved that if $r+s+t=0$\, then $rA + sA+tA$ contains a Bohr set (h
 ere $rA = \\{ ra: a \\in A \\}$). \nWe   investigate this phenomenon in mo
 re general groups $G$\, where $rA\, sA\, tA$ are replaced by images of $A$
  under certain endomomorphisms of $G$. It is also natural to ask for parti
 tion analogues of the Bergelson-Ruzsa theorem. In CANT 2021\, I discussed 
 our results in compact abelian groups (generalizations of $\\mathbf{R} /\\
 mathbf{Z}$). \\\nIn this talk\, I will discuss our progress on countable d
 iscrete abelian groups (generalizations of $\\mathbf{Z}$). The key ingredi
 ents are certain transference principles which allow us to transfer the re
 sults from compact groups to discrete countable groups. \nThis talk is bas
 ed on joint works with Anh Le\, and with Anh Le and John Griesmer.\n
LOCATION:https://researchseminars.org/talk/CANT2022/38/
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