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SUMMARY:Shunsuke Usuki (Kyoto University)
DTSTART:20240423T130000Z
DTEND:20240423T140000Z
DTSTAMP:20260423T021339Z
UID:OWNS/130
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/130/">O
 n a lower bound of the number of integers in Littlewood's conjecture</a>\n
 by Shunsuke Usuki (Kyoto University) as part of One World Numeration semin
 ar\n\n\nAbstract\nLittlewood's conjecture is a famous and long-standing op
 en problem which states that\, for every $(\\alpha\,\\beta) \\in \\mathbb{
 R}^2$\, $n\\|n\\alpha\\|\\|n\\beta\\|$ can be arbitrarily small for some i
 nteger $n$.\nThis problem is closely related to the action of diagonal mat
 rices on $\\mathrm{SL}(3\,\\mathbb{R})/\\mathrm{SL}(3\,\\mathbb{Z})$\, and
  a groundbreaking result was shown by Einsiedler\, Katok and Lindenstrauss
  from the measure rigidity for this action\, saying that Littlewood's conj
 ecture is true except on a set of Hausdorff dimension zero.\nIn this talk\
 , I will explain about a new quantitative result on Littlewood's conjectur
 e which gives\, for every $(\\alpha\,\\beta) \\in \\mathbb{R}^2$ except on
  sets of small Hausdorff dimension\, an estimate of the number of integers
  $n$ which make $n\\|n\\alpha\\|\\|n\\beta\\|$ small. The keys for the pro
 of are the measure rigidity and further studies on behavior of empirical m
 easures for the diagonal action.\n
LOCATION:https://researchseminars.org/talk/OWNS/130/
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